src/HOL/Library/Multiset.thy
author nipkow
Fri, 13 Feb 2009 23:55:04 +0100
changeset 29901 f4b3f8fbf599
parent 29509 1ff0f3f08a7b
child 30428 14f469e70eab
permissions -rw-r--r--
finiteness lemmas
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
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
     2
    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     3
*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     4
14706
71590b7733b7 tuned document;
wenzelm
parents: 14691
diff changeset
     5
header {* Multisets *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
     6
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
     7
theory Multiset
27487
c8a6ce181805 absolute imports of HOL/*.thy theories
haftmann
parents: 27368
diff changeset
     8
imports Plain "~~/src/HOL/List"
15131
c69542757a4d New theory header syntax.
nipkow
parents: 15072
diff changeset
     9
begin
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    10
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    11
subsection {* The type of multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    12
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25140
diff changeset
    13
typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    14
proof
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
    15
  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    16
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    17
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    18
lemmas multiset_typedef [simp] =
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
    19
    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
    20
  and [simp] = Rep_multiset_inject [symmetric]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    21
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    22
definition Mempty :: "'a multiset"  ("{#}") where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    23
  [code del]: "{#} = Abs_multiset (\<lambda>a. 0)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    24
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    25
definition single :: "'a => 'a multiset" where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    26
  [code del]: "single 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
    27
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    28
definition count :: "'a multiset => 'a => nat" where
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    29
  "count = Rep_multiset"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    30
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    31
definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    32
  "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
    33
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    34
abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
    35
  "a :# M == 0 < count M a"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
    36
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    37
notation (xsymbols)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    38
  Melem (infix "\<in>#" 50)
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
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
    41
  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    42
translations
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
    43
  "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    44
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    45
definition set_of :: "'a multiset => 'a set" where
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    46
  "set_of M = {x. x :# M}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    47
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    48
instantiation multiset :: (type) "{plus, minus, zero, size}" 
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    49
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    50
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    51
definition union_def [code del]:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    52
  "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    53
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    54
definition diff_def [code del]:
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    55
  "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    56
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    57
definition Zero_multiset_def [simp]:
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    58
  "0 = {#}"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    59
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    60
definition size_def:
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    61
  "size M = setsum (count M) (set_of M)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    62
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    63
instance ..
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    64
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    65
end
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    66
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    67
definition
21404
eb85850d3eb7 more robust syntax for definition/abbreviation/notation;
wenzelm
parents: 21214
diff changeset
    68
  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    69
  "multiset_inter A B = A - (A - B)"
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
    70
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    71
text {* Multiset Enumeration *}
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    72
syntax
26176
038baad81209 tuned syntax declaration;
wenzelm
parents: 26145
diff changeset
    73
  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
25507
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
    74
translations
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
    75
  "{#x, xs#}" == "{#x#} + {#xs#}"
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
    76
  "{#x#}" == "CONST single x"
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
    77
10249
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
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    80
 \medskip Preservation of the representing set @{term multiset}.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    81
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    82
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
    83
lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
26178
nipkow
parents: 26176
diff changeset
    84
by (simp add: multiset_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    85
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
    86
lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
26178
nipkow
parents: 26176
diff changeset
    87
by (simp add: multiset_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    88
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
    89
lemma union_preserves_multiset:
26178
nipkow
parents: 26176
diff changeset
    90
  "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29509
diff changeset
    91
by (simp add: multiset_def)
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29509
diff changeset
    92
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    93
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
    94
lemma diff_preserves_multiset:
26178
nipkow
parents: 26176
diff changeset
    95
  "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
nipkow
parents: 26176
diff changeset
    96
apply (simp add: multiset_def)
nipkow
parents: 26176
diff changeset
    97
apply (rule finite_subset)
nipkow
parents: 26176
diff changeset
    98
 apply auto
nipkow
parents: 26176
diff changeset
    99
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   100
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   101
lemma MCollect_preserves_multiset:
26178
nipkow
parents: 26176
diff changeset
   102
  "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
nipkow
parents: 26176
diff changeset
   103
apply (simp add: multiset_def)
nipkow
parents: 26176
diff changeset
   104
apply (rule finite_subset, auto)
nipkow
parents: 26176
diff changeset
   105
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   106
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   107
lemmas in_multiset = const0_in_multiset only1_in_multiset
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   108
  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   109
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   110
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   111
subsection {* Algebraic properties *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   112
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   113
subsubsection {* Union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   114
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   115
lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
26178
nipkow
parents: 26176
diff changeset
   116
by (simp add: union_def Mempty_def in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   117
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   118
lemma union_commute: "M + N = N + (M::'a multiset)"
26178
nipkow
parents: 26176
diff changeset
   119
by (simp add: union_def add_ac in_multiset)
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   120
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   121
lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
26178
nipkow
parents: 26176
diff changeset
   122
by (simp add: union_def add_ac in_multiset)
10249
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 union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   125
proof -
26178
nipkow
parents: 26176
diff changeset
   126
  have "M + (N + K) = (N + K) + M" by (rule union_commute)
nipkow
parents: 26176
diff changeset
   127
  also have "\<dots> = N + (K + M)" by (rule union_assoc)
nipkow
parents: 26176
diff changeset
   128
  also have "K + M = M + K" by (rule union_commute)
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   129
  finally show ?thesis .
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   130
qed
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   131
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   132
lemmas union_ac = union_assoc union_commute union_lcomm
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   133
14738
83f1a514dcb4 changes made due to new Ring_and_Field theory
obua
parents: 14722
diff changeset
   134
instance multiset :: (type) comm_monoid_add
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   135
proof
14722
8e739a6eaf11 replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
obua
parents: 14706
diff changeset
   136
  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
   137
  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
   138
  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
   139
  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
   140
qed
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   141
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   142
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   143
subsubsection {* Difference *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   144
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   145
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
26178
nipkow
parents: 26176
diff changeset
   146
by (simp add: Mempty_def diff_def in_multiset)
10249
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 diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
26178
nipkow
parents: 26176
diff changeset
   149
by (simp add: union_def diff_def in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   150
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   151
lemma diff_cancel: "A - A = {#}"
26178
nipkow
parents: 26176
diff changeset
   152
by (simp add: diff_def Mempty_def)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   153
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   154
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   155
subsubsection {* Count of elements *}
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 count_empty [simp]: "count {#} a = 0"
26178
nipkow
parents: 26176
diff changeset
   158
by (simp add: count_def Mempty_def in_multiset)
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 count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
26178
nipkow
parents: 26176
diff changeset
   161
by (simp add: count_def single_def in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   162
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   163
lemma count_union [simp]: "count (M + N) a = count M a + count N a"
26178
nipkow
parents: 26176
diff changeset
   164
by (simp add: count_def union_def in_multiset)
10249
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 count_diff [simp]: "count (M - N) a = count M a - count N a"
26178
nipkow
parents: 26176
diff changeset
   167
by (simp add: count_def diff_def in_multiset)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   168
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   169
lemma count_MCollect [simp]:
26178
nipkow
parents: 26176
diff changeset
   170
  "count {# x:#M. P x #} a = (if P a then count M a else 0)"
nipkow
parents: 26176
diff changeset
   171
by (simp add: count_def MCollect_def in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   172
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   173
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   174
subsubsection {* Set of elements *}
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 set_of_empty [simp]: "set_of {#} = {}"
26178
nipkow
parents: 26176
diff changeset
   177
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   178
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   179
lemma set_of_single [simp]: "set_of {#b#} = {b}"
26178
nipkow
parents: 26176
diff changeset
   180
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   181
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   182
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
26178
nipkow
parents: 26176
diff changeset
   183
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   184
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   185
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
26818
b4a24433154e Instantiated rule expand_fun_eq in proof of set_of_eq_empty_iff, to avoid that
berghofe
parents: 26567
diff changeset
   186
by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   187
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   188
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
26178
nipkow
parents: 26176
diff changeset
   189
by (auto simp add: set_of_def)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   190
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   191
lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
26178
nipkow
parents: 26176
diff changeset
   192
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   193
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   194
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   195
subsubsection {* Size *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   196
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   197
lemma size_empty [simp]: "size {#} = 0"
26178
nipkow
parents: 26176
diff changeset
   198
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   199
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   200
lemma size_single [simp]: "size {#b#} = 1"
26178
nipkow
parents: 26176
diff changeset
   201
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   202
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   203
lemma finite_set_of [iff]: "finite (set_of M)"
26178
nipkow
parents: 26176
diff changeset
   204
using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
10249
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 setsum_count_Int:
26178
nipkow
parents: 26176
diff changeset
   207
  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow
parents: 26176
diff changeset
   208
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   209
 apply simp
nipkow
parents: 26176
diff changeset
   210
apply (simp add: Int_insert_left set_of_def)
nipkow
parents: 26176
diff changeset
   211
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   212
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   213
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
26178
nipkow
parents: 26176
diff changeset
   214
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   215
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow
parents: 26176
diff changeset
   216
 prefer 2
nipkow
parents: 26176
diff changeset
   217
 apply (rule ext, simp)
nipkow
parents: 26176
diff changeset
   218
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow
parents: 26176
diff changeset
   219
apply (subst Int_commute)
nipkow
parents: 26176
diff changeset
   220
apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow
parents: 26176
diff changeset
   221
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   222
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   223
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
26178
nipkow
parents: 26176
diff changeset
   224
apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
nipkow
parents: 26176
diff changeset
   225
apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
nipkow
parents: 26176
diff changeset
   226
done
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   227
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   228
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
26178
nipkow
parents: 26176
diff changeset
   229
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   230
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   231
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
26178
nipkow
parents: 26176
diff changeset
   232
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   233
apply (drule setsum_SucD)
nipkow
parents: 26176
diff changeset
   234
apply auto
nipkow
parents: 26176
diff changeset
   235
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   236
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   237
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   238
subsubsection {* Equality of multisets *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   239
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   240
lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
26178
nipkow
parents: 26176
diff changeset
   241
by (simp add: count_def expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   242
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   243
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
26178
nipkow
parents: 26176
diff changeset
   244
by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   245
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   246
lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
26178
nipkow
parents: 26176
diff changeset
   247
by (auto simp add: single_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   248
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   249
lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
26178
nipkow
parents: 26176
diff changeset
   250
by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   251
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   252
lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
26178
nipkow
parents: 26176
diff changeset
   253
by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   254
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   255
lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
26178
nipkow
parents: 26176
diff changeset
   256
by (simp add: union_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   257
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   258
lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
26178
nipkow
parents: 26176
diff changeset
   259
by (simp add: union_def in_multiset expand_fun_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   260
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   261
lemma union_is_single:
26178
nipkow
parents: 26176
diff changeset
   262
  "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
nipkow
parents: 26176
diff changeset
   263
apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
nipkow
parents: 26176
diff changeset
   264
apply blast
nipkow
parents: 26176
diff changeset
   265
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   266
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   267
lemma single_is_union:
26178
nipkow
parents: 26176
diff changeset
   268
  "({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
nipkow
parents: 26176
diff changeset
   269
apply (unfold Mempty_def single_def union_def)
nipkow
parents: 26176
diff changeset
   270
apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
nipkow
parents: 26176
diff changeset
   271
apply (blast dest: sym)
nipkow
parents: 26176
diff changeset
   272
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   273
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   274
lemma add_eq_conv_diff:
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   275
  "(M + {#a#} = N + {#b#}) =
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   276
   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
26178
nipkow
parents: 26176
diff changeset
   277
using [[simproc del: neq]]
nipkow
parents: 26176
diff changeset
   278
apply (unfold single_def union_def diff_def)
nipkow
parents: 26176
diff changeset
   279
apply (simp (no_asm) add: in_multiset expand_fun_eq)
nipkow
parents: 26176
diff changeset
   280
apply (rule conjI, force, safe, simp_all)
nipkow
parents: 26176
diff changeset
   281
apply (simp add: eq_sym_conv)
nipkow
parents: 26176
diff changeset
   282
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   283
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   284
declare Rep_multiset_inject [symmetric, simp del]
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   285
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   286
instance multiset :: (type) cancel_ab_semigroup_add
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   287
proof
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   288
  fix a b c :: "'a multiset"
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   289
  show "a + b = a + c \<Longrightarrow> b = c" by simp
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   290
qed
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   291
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   292
lemma insert_DiffM:
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   293
  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
26178
nipkow
parents: 26176
diff changeset
   294
by (clarsimp simp: multiset_eq_conv_count_eq)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   295
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   296
lemma insert_DiffM2[simp]:
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   297
  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
26178
nipkow
parents: 26176
diff changeset
   298
by (clarsimp simp: multiset_eq_conv_count_eq)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   299
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   300
lemma multi_union_self_other_eq: 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   301
  "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
26178
nipkow
parents: 26176
diff changeset
   302
by (induct A arbitrary: X Y) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   303
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   304
lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
26178
nipkow
parents: 26176
diff changeset
   305
by (metis single_not_empty union_empty union_left_cancel)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   306
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   307
lemma insert_noteq_member: 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   308
  assumes BC: "B + {#b#} = C + {#c#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   309
   and bnotc: "b \<noteq> c"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   310
  shows "c \<in># B"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   311
proof -
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   312
  have "c \<in># C + {#c#}" by simp
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   313
  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   314
  then have "c \<in># B + {#b#}" using BC by simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   315
  then show "c \<in># B" using nc by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   316
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   317
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   318
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   319
lemma add_eq_conv_ex:
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   320
  "(M + {#a#} = N + {#b#}) =
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   321
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
26178
nipkow
parents: 26176
diff changeset
   322
by (auto simp add: add_eq_conv_diff)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   323
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   324
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   325
lemma empty_multiset_count:
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   326
  "(\<forall>x. count A x = 0) = (A = {#})"
26178
nipkow
parents: 26176
diff changeset
   327
by (metis count_empty multiset_eq_conv_count_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   328
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   329
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   330
subsubsection {* Intersection *}
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   331
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   332
lemma multiset_inter_count:
26178
nipkow
parents: 26176
diff changeset
   333
  "count (A #\<inter> B) x = min (count A x) (count B x)"
nipkow
parents: 26176
diff changeset
   334
by (simp add: multiset_inter_def min_def)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   335
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   336
lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
26178
nipkow
parents: 26176
diff changeset
   337
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
21214
a91bab12b2bd adjusted two lemma names due to name change in interpretation
haftmann
parents: 20770
diff changeset
   338
    min_max.inf_commute)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   339
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   340
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
26178
nipkow
parents: 26176
diff changeset
   341
by (simp add: multiset_eq_conv_count_eq multiset_inter_count
21214
a91bab12b2bd adjusted two lemma names due to name change in interpretation
haftmann
parents: 20770
diff changeset
   342
    min_max.inf_assoc)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   343
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   344
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
26178
nipkow
parents: 26176
diff changeset
   345
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   346
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   347
lemmas multiset_inter_ac =
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   348
  multiset_inter_commute
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   349
  multiset_inter_assoc
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   350
  multiset_inter_left_commute
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   351
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   352
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
26178
nipkow
parents: 26176
diff changeset
   353
by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   354
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   355
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
26178
nipkow
parents: 26176
diff changeset
   356
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   357
    split: split_if_asm)
26178
nipkow
parents: 26176
diff changeset
   358
apply clarsimp
nipkow
parents: 26176
diff changeset
   359
apply (erule_tac x = a in allE)
nipkow
parents: 26176
diff changeset
   360
apply auto
nipkow
parents: 26176
diff changeset
   361
done
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   362
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   363
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   364
subsubsection {* Comprehension (filter) *}
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   365
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   366
lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
26178
nipkow
parents: 26176
diff changeset
   367
by (simp add: MCollect_def Mempty_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   368
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   369
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   370
lemma MCollect_single [simp]:
26178
nipkow
parents: 26176
diff changeset
   371
  "MCollect {#x#} P = (if P x then {#x#} else {#})"
nipkow
parents: 26176
diff changeset
   372
by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   373
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   374
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   375
lemma MCollect_union [simp]:
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   376
  "MCollect (M+N) f = MCollect M f + MCollect N f"
26178
nipkow
parents: 26176
diff changeset
   377
by (simp add: MCollect_def union_def Abs_multiset_inject
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   378
    in_multiset expand_fun_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   379
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   380
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   381
subsection {* Induction and case splits *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   382
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   383
lemma setsum_decr:
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   384
  "finite F ==> (0::nat) < f a ==>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   385
    setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
26178
nipkow
parents: 26176
diff changeset
   386
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   387
 apply auto
nipkow
parents: 26176
diff changeset
   388
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow
parents: 26176
diff changeset
   389
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   390
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   391
lemma rep_multiset_induct_aux:
26178
nipkow
parents: 26176
diff changeset
   392
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow
parents: 26176
diff changeset
   393
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow
parents: 26176
diff changeset
   394
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow
parents: 26176
diff changeset
   395
apply (unfold multiset_def)
nipkow
parents: 26176
diff changeset
   396
apply (induct_tac n, simp, clarify)
nipkow
parents: 26176
diff changeset
   397
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow
parents: 26176
diff changeset
   398
  apply simp
nipkow
parents: 26176
diff changeset
   399
  apply (rule 1)
nipkow
parents: 26176
diff changeset
   400
 apply (rule ext, force, clarify)
nipkow
parents: 26176
diff changeset
   401
apply (frule setsum_SucD, clarify)
nipkow
parents: 26176
diff changeset
   402
apply (rename_tac a)
nipkow
parents: 26176
diff changeset
   403
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow
parents: 26176
diff changeset
   404
 prefer 2
nipkow
parents: 26176
diff changeset
   405
 apply (rule finite_subset)
nipkow
parents: 26176
diff changeset
   406
  prefer 2
nipkow
parents: 26176
diff changeset
   407
  apply assumption
nipkow
parents: 26176
diff changeset
   408
 apply simp
nipkow
parents: 26176
diff changeset
   409
 apply blast
nipkow
parents: 26176
diff changeset
   410
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow
parents: 26176
diff changeset
   411
 prefer 2
nipkow
parents: 26176
diff changeset
   412
 apply (rule ext)
nipkow
parents: 26176
diff changeset
   413
 apply (simp (no_asm_simp))
nipkow
parents: 26176
diff changeset
   414
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow
parents: 26176
diff changeset
   415
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow
parents: 26176
diff changeset
   416
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow
parents: 26176
diff changeset
   417
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow
parents: 26176
diff changeset
   418
 prefer 2
nipkow
parents: 26176
diff changeset
   419
 apply blast
nipkow
parents: 26176
diff changeset
   420
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow
parents: 26176
diff changeset
   421
 prefer 2
nipkow
parents: 26176
diff changeset
   422
 apply blast
nipkow
parents: 26176
diff changeset
   423
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow
parents: 26176
diff changeset
   424
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   425
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   426
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   427
  "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
   428
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
26178
nipkow
parents: 26176
diff changeset
   429
using rep_multiset_induct_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   430
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   431
theorem multiset_induct [case_names empty add, induct type: multiset]:
26178
nipkow
parents: 26176
diff changeset
   432
assumes empty: "P {#}"
nipkow
parents: 26176
diff changeset
   433
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow
parents: 26176
diff changeset
   434
shows "P M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   435
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   436
  note defns = union_def single_def Mempty_def
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   437
  show ?thesis
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   438
    apply (rule Rep_multiset_inverse [THEN subst])
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   439
    apply (rule Rep_multiset [THEN rep_multiset_induct])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   440
     apply (rule empty [unfolded defns])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   441
    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
   442
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   443
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   444
    apply (erule ssubst)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   445
    apply (erule Abs_multiset_inverse [THEN subst])
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   446
    apply (drule add [unfolded defns, simplified])
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   447
    apply(simp add:in_multiset)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   448
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   449
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   450
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   451
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   452
by (induct M) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   453
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   454
lemma multiset_cases [cases type, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
   455
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   456
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   457
shows "P"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   458
proof (cases "M = {#}")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   459
  assume "M = {#}" then show ?thesis using em by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   460
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   461
  assume "M \<noteq> {#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   462
  then obtain M' m where "M = M' + {#m#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   463
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   464
  then show ?thesis using add by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   465
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   466
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   467
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
   468
apply (cases M)
nipkow
parents: 26176
diff changeset
   469
 apply simp
nipkow
parents: 26176
diff changeset
   470
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow
parents: 26176
diff changeset
   471
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   472
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   473
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
26178
nipkow
parents: 26176
diff changeset
   474
apply (subst multiset_eq_conv_count_eq)
nipkow
parents: 26176
diff changeset
   475
apply auto
nipkow
parents: 26176
diff changeset
   476
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   477
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   478
declare multiset_typedef [simp del]
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   479
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   480
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
26178
nipkow
parents: 26176
diff changeset
   481
by (cases "B = {#}") (auto dest: multi_member_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   482
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   483
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   484
subsection {* Orderings *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   485
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   486
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   487
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   488
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   489
  [code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   490
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   491
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   492
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   493
  "mult r = (mult1 r)\<^sup>+"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   494
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   495
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
26178
nipkow
parents: 26176
diff changeset
   496
by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   497
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   498
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   499
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   500
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 19564
diff changeset
   501
  (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   502
proof (unfold mult1_def)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   503
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   504
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   505
  let ?case1 = "?case1 {(N, M). ?R N M}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   506
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   507
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   508
  then have "\<exists>a' M0' K.
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   509
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   510
  then show "?case1 \<or> ?case2"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   511
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   512
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   513
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   514
    assume "M0 + {#a#} = M0' + {#a'#}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   515
    then have "M0 = M0' \<and> a = a' \<or>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   516
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   517
      by (simp only: add_eq_conv_ex)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   518
    then show ?thesis
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   519
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   520
      assume "M0 = M0'" "a = a'"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   521
      with N r have "?r K a \<and> N = M0 + K" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   522
      then have ?case2 .. then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   523
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   524
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   525
      assume "M0' = K' + {#a#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   526
      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
   527
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   528
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
      with r have "?R (K' + K) M0" by blast
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   530
      with n have ?case1 by simp then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   531
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   532
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   533
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   534
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   535
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
   536
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   537
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   538
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   539
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   540
    fix M M0 a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   541
    assume M0: "M0 \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   542
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   543
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   544
    have "M0 + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   545
    proof (rule accI [of "M0 + {#a#}"])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   546
      fix N
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   547
      assume "(N, M0 + {#a#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   548
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   549
          (\<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
   550
        by (rule less_add)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   551
      then show "N \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   552
      proof (elim exE disjE conjE)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   553
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   554
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   555
        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   556
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   557
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   558
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   559
        assume N: "N = M0 + K"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   560
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   561
        then have "M0 + K \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   562
        proof (induct K)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   563
          case empty
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   564
          from M0 show "M0 + {#} \<in> ?W" by simp
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   565
        next
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   566
          case (add K x)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   567
          from add.prems have "(x, a) \<in> r" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   568
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   569
          moreover from add have "M0 + K \<in> ?W" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   570
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   571
          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
   572
        qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   573
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   574
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   575
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   576
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   577
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   578
  assume wf: "wf r"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   579
  fix M
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   580
  show "M \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   581
  proof (induct M)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   582
    show "{#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   583
    proof (rule accI)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   584
      fix b assume "(b, {#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   585
      with not_less_empty show "b \<in> ?W" by contradiction
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   586
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   587
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   588
    fix M a assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   589
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   590
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   591
      fix a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   592
      assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   593
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   594
      proof
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   595
        fix M assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   596
        then show "M + {#a#} \<in> ?W"
23373
ead82c82da9e tuned proofs: avoid implicit prems;
wenzelm
parents: 23281
diff changeset
   597
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   598
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   599
    qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   600
    from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   601
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   602
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   603
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   604
theorem wf_mult1: "wf r ==> wf (mult1 r)"
26178
nipkow
parents: 26176
diff changeset
   605
by (rule acc_wfI) (rule all_accessible)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   606
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   607
theorem wf_mult: "wf r ==> wf (mult r)"
26178
nipkow
parents: 26176
diff changeset
   608
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   609
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   610
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   611
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   612
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   613
(*Badly needed: a linear arithmetic procedure for multisets*)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   615
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
26178
nipkow
parents: 26176
diff changeset
   616
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   617
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   618
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   619
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   620
lemma mult_implies_one_step:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   621
  "trans r ==> (M, N) \<in> mult r ==>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   622
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   623
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
26178
nipkow
parents: 26176
diff changeset
   624
apply (unfold mult_def mult1_def set_of_def)
nipkow
parents: 26176
diff changeset
   625
apply (erule converse_trancl_induct, clarify)
nipkow
parents: 26176
diff changeset
   626
 apply (rule_tac x = M0 in exI, simp, clarify)
nipkow
parents: 26176
diff changeset
   627
apply (case_tac "a :# K")
nipkow
parents: 26176
diff changeset
   628
 apply (rule_tac x = I in exI)
nipkow
parents: 26176
diff changeset
   629
 apply (simp (no_asm))
nipkow
parents: 26176
diff changeset
   630
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
nipkow
parents: 26176
diff changeset
   631
 apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow
parents: 26176
diff changeset
   632
 apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow
parents: 26176
diff changeset
   633
 apply (simp add: diff_union_single_conv)
nipkow
parents: 26176
diff changeset
   634
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
   635
 apply blast
nipkow
parents: 26176
diff changeset
   636
apply (subgoal_tac "a :# I")
nipkow
parents: 26176
diff changeset
   637
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow
parents: 26176
diff changeset
   638
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow
parents: 26176
diff changeset
   639
 apply (rule_tac x = "K + Ka" in exI)
nipkow
parents: 26176
diff changeset
   640
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
   641
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
   642
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
   643
  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow
parents: 26176
diff changeset
   644
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
   645
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
   646
 apply blast
nipkow
parents: 26176
diff changeset
   647
apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow
parents: 26176
diff changeset
   648
 apply simp
nipkow
parents: 26176
diff changeset
   649
apply (simp (no_asm))
nipkow
parents: 26176
diff changeset
   650
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   651
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   652
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   653
by (simp add: multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   654
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   655
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   656
apply (erule size_eq_Suc_imp_elem [THEN exE])
nipkow
parents: 26176
diff changeset
   657
apply (drule elem_imp_eq_diff_union, auto)
nipkow
parents: 26176
diff changeset
   658
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   659
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   660
lemma one_step_implies_mult_aux:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   661
  "trans r ==>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   662
    \<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))
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   663
      --> (I + K, I + J) \<in> mult r"
26178
nipkow
parents: 26176
diff changeset
   664
apply (induct_tac n, auto)
nipkow
parents: 26176
diff changeset
   665
apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow
parents: 26176
diff changeset
   666
apply (rename_tac "J'", simp)
nipkow
parents: 26176
diff changeset
   667
apply (erule notE, auto)
nipkow
parents: 26176
diff changeset
   668
apply (case_tac "J' = {#}")
nipkow
parents: 26176
diff changeset
   669
 apply (simp add: mult_def)
nipkow
parents: 26176
diff changeset
   670
 apply (rule r_into_trancl)
nipkow
parents: 26176
diff changeset
   671
 apply (simp add: mult1_def set_of_def, blast)
nipkow
parents: 26176
diff changeset
   672
txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow
parents: 26176
diff changeset
   673
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow
parents: 26176
diff changeset
   674
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow
parents: 26176
diff changeset
   675
apply (erule ssubst)
nipkow
parents: 26176
diff changeset
   676
apply (simp add: Ball_def, auto)
nipkow
parents: 26176
diff changeset
   677
apply (subgoal_tac
nipkow
parents: 26176
diff changeset
   678
  "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow
parents: 26176
diff changeset
   679
    (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow
parents: 26176
diff changeset
   680
 prefer 2
nipkow
parents: 26176
diff changeset
   681
 apply force
nipkow
parents: 26176
diff changeset
   682
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
nipkow
parents: 26176
diff changeset
   683
apply (erule trancl_trans)
nipkow
parents: 26176
diff changeset
   684
apply (rule r_into_trancl)
nipkow
parents: 26176
diff changeset
   685
apply (simp add: mult1_def set_of_def)
nipkow
parents: 26176
diff changeset
   686
apply (rule_tac x = a in exI)
nipkow
parents: 26176
diff changeset
   687
apply (rule_tac x = "I + J'" in exI)
nipkow
parents: 26176
diff changeset
   688
apply (simp add: union_ac)
nipkow
parents: 26176
diff changeset
   689
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   690
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   691
lemma one_step_implies_mult:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   692
  "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   693
    ==> (I + K, I + J) \<in> mult r"
26178
nipkow
parents: 26176
diff changeset
   694
using one_step_implies_mult_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   695
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   696
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   697
subsubsection {* Partial-order properties *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   698
26567
7bcebb8c2d33 instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents: 26178
diff changeset
   699
instantiation multiset :: (order) order
7bcebb8c2d33 instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents: 26178
diff changeset
   700
begin
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   701
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   702
definition less_multiset_def [code del]:
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   703
  "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
26567
7bcebb8c2d33 instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents: 26178
diff changeset
   704
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   705
definition le_multiset_def [code del]:
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   706
  "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   707
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
   708
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
26178
nipkow
parents: 26176
diff changeset
   709
unfolding trans_def by (blast intro: order_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   710
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   711
text {*
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   712
 \medskip Irreflexivity.
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   713
*}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   714
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   715
lemma mult_irrefl_aux:
26178
nipkow
parents: 26176
diff changeset
   716
  "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
nipkow
parents: 26176
diff changeset
   717
by (induct rule: finite_induct) (auto intro: order_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   718
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   719
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   720
apply (unfold less_multiset_def, auto)
nipkow
parents: 26176
diff changeset
   721
apply (drule trans_base_order [THEN mult_implies_one_step], auto)
nipkow
parents: 26176
diff changeset
   722
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
nipkow
parents: 26176
diff changeset
   723
apply (simp add: set_of_eq_empty_iff)
nipkow
parents: 26176
diff changeset
   724
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   725
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   726
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
26178
nipkow
parents: 26176
diff changeset
   727
using insert mult_less_not_refl by fast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   728
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   729
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   730
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   731
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   732
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   733
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   734
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   735
text {* Asymmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   736
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   737
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   738
apply auto
nipkow
parents: 26176
diff changeset
   739
apply (rule mult_less_not_refl [THEN notE])
nipkow
parents: 26176
diff changeset
   740
apply (erule mult_less_trans, assumption)
nipkow
parents: 26176
diff changeset
   741
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   742
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   743
theorem mult_less_asym:
26178
nipkow
parents: 26176
diff changeset
   744
  "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
nipkow
parents: 26176
diff changeset
   745
using mult_less_not_sym by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   746
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   747
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   748
unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   749
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   750
text {* Anti-symmetry. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   751
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   752
theorem mult_le_antisym:
26178
nipkow
parents: 26176
diff changeset
   753
  "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   754
unfolding le_multiset_def by (blast dest: mult_less_not_sym)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   755
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   756
text {* Transitivity. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   757
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   758
theorem mult_le_trans:
26178
nipkow
parents: 26176
diff changeset
   759
  "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   760
unfolding le_multiset_def by (blast intro: mult_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   761
11655
923e4d0d36d5 tuned parentheses in relational expressions;
wenzelm
parents: 11549
diff changeset
   762
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
26178
nipkow
parents: 26176
diff changeset
   763
unfolding le_multiset_def by auto
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   764
27682
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
   765
instance proof
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
   766
qed (auto simp add: mult_less_le dest: mult_le_antisym elim: mult_le_trans)
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   767
26567
7bcebb8c2d33 instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents: 26178
diff changeset
   768
end
7bcebb8c2d33 instantiation replacing primitive instance plus overloaded defs; more conservative type arities
haftmann
parents: 26178
diff changeset
   769
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   770
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   771
subsubsection {* Monotonicity of multiset union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   772
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   773
lemma mult1_union:
26178
nipkow
parents: 26176
diff changeset
   774
  "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
nipkow
parents: 26176
diff changeset
   775
apply (unfold mult1_def)
nipkow
parents: 26176
diff changeset
   776
apply auto
nipkow
parents: 26176
diff changeset
   777
apply (rule_tac x = a in exI)
nipkow
parents: 26176
diff changeset
   778
apply (rule_tac x = "C + M0" in exI)
nipkow
parents: 26176
diff changeset
   779
apply (simp add: union_assoc)
nipkow
parents: 26176
diff changeset
   780
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   781
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   782
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   783
apply (unfold less_multiset_def mult_def)
nipkow
parents: 26176
diff changeset
   784
apply (erule trancl_induct)
nipkow
parents: 26176
diff changeset
   785
 apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
nipkow
parents: 26176
diff changeset
   786
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
nipkow
parents: 26176
diff changeset
   787
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   788
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   789
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   790
apply (subst union_commute [of B C])
nipkow
parents: 26176
diff changeset
   791
apply (subst union_commute [of D C])
nipkow
parents: 26176
diff changeset
   792
apply (erule union_less_mono2)
nipkow
parents: 26176
diff changeset
   793
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   794
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   795
lemma union_less_mono:
26178
nipkow
parents: 26176
diff changeset
   796
  "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   797
by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   798
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   799
lemma union_le_mono:
26178
nipkow
parents: 26176
diff changeset
   800
  "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
nipkow
parents: 26176
diff changeset
   801
unfolding le_multiset_def
nipkow
parents: 26176
diff changeset
   802
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
   803
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   804
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   805
apply (unfold le_multiset_def less_multiset_def)
nipkow
parents: 26176
diff changeset
   806
apply (case_tac "M = {#}")
nipkow
parents: 26176
diff changeset
   807
 prefer 2
nipkow
parents: 26176
diff changeset
   808
 apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
nipkow
parents: 26176
diff changeset
   809
  prefer 2
nipkow
parents: 26176
diff changeset
   810
  apply (rule one_step_implies_mult)
nipkow
parents: 26176
diff changeset
   811
    apply (simp only: trans_def)
nipkow
parents: 26176
diff changeset
   812
    apply auto
nipkow
parents: 26176
diff changeset
   813
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   814
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   815
lemma union_upper1: "A <= A + (B::'a::order multiset)"
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   816
proof -
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   817
  have "A + {#} <= A + B" by (blast intro: union_le_mono)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   818
  then show ?thesis by simp
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   819
qed
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   820
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   821
lemma union_upper2: "B <= A + (B::'a::order multiset)"
26178
nipkow
parents: 26176
diff changeset
   822
by (subst union_commute) (rule union_upper1)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   823
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   824
instance multiset :: (order) pordered_ab_semigroup_add
26178
nipkow
parents: 26176
diff changeset
   825
apply intro_classes
nipkow
parents: 26176
diff changeset
   826
apply (erule union_le_mono[OF mult_le_refl])
nipkow
parents: 26176
diff changeset
   827
done
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   828
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   829
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   830
subsection {* Link with lists *}
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   831
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   832
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   833
  "multiset_of [] = {#}" |
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   834
  "multiset_of (a # x) = multiset_of x + {# a #}"
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 multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
26178
nipkow
parents: 26176
diff changeset
   837
by (induct x) 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_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
26178
nipkow
parents: 26176
diff changeset
   840
by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   841
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   842
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
26178
nipkow
parents: 26176
diff changeset
   843
by (induct x) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   844
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   845
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
26178
nipkow
parents: 26176
diff changeset
   846
by (induct xs) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   847
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   848
lemma multiset_of_append [simp]:
26178
nipkow
parents: 26176
diff changeset
   849
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
nipkow
parents: 26176
diff changeset
   850
by (induct xs arbitrary: ys) (auto simp: union_ac)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
   851
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   852
lemma surj_multiset_of: "surj multiset_of"
26178
nipkow
parents: 26176
diff changeset
   853
apply (unfold surj_def)
nipkow
parents: 26176
diff changeset
   854
apply (rule allI)
nipkow
parents: 26176
diff changeset
   855
apply (rule_tac M = y in multiset_induct)
nipkow
parents: 26176
diff changeset
   856
 apply auto
nipkow
parents: 26176
diff changeset
   857
apply (rule_tac x = "x # xa" in exI)
nipkow
parents: 26176
diff changeset
   858
apply auto
nipkow
parents: 26176
diff changeset
   859
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   860
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25140
diff changeset
   861
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
26178
nipkow
parents: 26176
diff changeset
   862
by (induct x) auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   863
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   864
lemma distinct_count_atmost_1:
26178
nipkow
parents: 26176
diff changeset
   865
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
nipkow
parents: 26176
diff changeset
   866
apply (induct x, simp, rule iffI, simp_all)
nipkow
parents: 26176
diff changeset
   867
apply (rule conjI)
nipkow
parents: 26176
diff changeset
   868
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
nipkow
parents: 26176
diff changeset
   869
apply (erule_tac x = a in allE, simp, clarify)
nipkow
parents: 26176
diff changeset
   870
apply (erule_tac x = aa in allE, simp)
nipkow
parents: 26176
diff changeset
   871
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   872
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   873
lemma multiset_of_eq_setD:
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   874
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
26178
nipkow
parents: 26176
diff changeset
   875
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   876
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   877
lemma set_eq_iff_multiset_of_eq_distinct:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   878
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   879
    (set x = set y) = (multiset_of x = multiset_of y)"
26178
nipkow
parents: 26176
diff changeset
   880
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
   881
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   882
lemma set_eq_iff_multiset_of_remdups_eq:
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   883
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
26178
nipkow
parents: 26176
diff changeset
   884
apply (rule iffI)
nipkow
parents: 26176
diff changeset
   885
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
nipkow
parents: 26176
diff changeset
   886
apply (drule distinct_remdups [THEN distinct_remdups
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   887
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
26178
nipkow
parents: 26176
diff changeset
   888
apply simp
nipkow
parents: 26176
diff changeset
   889
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   890
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   891
lemma multiset_of_compl_union [simp]:
26178
nipkow
parents: 26176
diff changeset
   892
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
nipkow
parents: 26176
diff changeset
   893
by (induct xs) (auto simp: union_ac)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   894
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   895
lemma count_filter:
26178
nipkow
parents: 26176
diff changeset
   896
  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
nipkow
parents: 26176
diff changeset
   897
by (induct xs) auto
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   898
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   899
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
26178
nipkow
parents: 26176
diff changeset
   900
apply (induct ls arbitrary: i)
nipkow
parents: 26176
diff changeset
   901
 apply simp
nipkow
parents: 26176
diff changeset
   902
apply (case_tac i)
nipkow
parents: 26176
diff changeset
   903
 apply auto
nipkow
parents: 26176
diff changeset
   904
done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   905
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   906
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
26178
nipkow
parents: 26176
diff changeset
   907
by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   908
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   909
lemma multiset_of_eq_length:
26178
nipkow
parents: 26176
diff changeset
   910
assumes "multiset_of xs = multiset_of ys"
nipkow
parents: 26176
diff changeset
   911
shows "length xs = length ys"
nipkow
parents: 26176
diff changeset
   912
using assms
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   913
proof (induct arbitrary: ys rule: length_induct)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   914
  case (1 xs ys)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   915
  show ?case
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   916
  proof (cases xs)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   917
    case Nil with "1.prems" show ?thesis by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   918
  next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   919
    case (Cons x xs')
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   920
    note xCons = Cons
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   921
    show ?thesis
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   922
    proof (cases ys)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   923
      case Nil
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   924
      with "1.prems" Cons show ?thesis by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   925
    next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   926
      case (Cons y ys')
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   927
      have x_in_ys: "x = y \<or> x \<in> set ys'"
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   928
      proof (cases "x = y")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   929
	case True then show ?thesis ..
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   930
      next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   931
	case False
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   932
	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   933
	with False show ?thesis by (simp add: mem_set_multiset_eq)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   934
      qed
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   935
      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   936
	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   937
      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   938
	apply -
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   939
	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   940
	apply fastsimp
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   941
	done
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   942
      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   943
      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   944
      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   945
    qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   946
  qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   947
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   948
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   949
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   950
  This lemma shows which properties suffice to show that a function
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   951
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   952
*}
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   953
lemma properties_for_sort:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   954
  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   955
proof (induct xs arbitrary: ys)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   956
  case Nil then show ?case by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   957
next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   958
  case (Cons x xs)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   959
  then have "x \<in> set ys"
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   960
    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   961
  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   962
    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   963
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   964
15867
5c63b6c2f4a5 some more lemmas about multiset_of
kleing
parents: 15630
diff changeset
   965
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   966
subsection {* Pointwise ordering induced by count *}
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   967
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   968
definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   969
  [code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   970
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   971
definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   972
  [code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   973
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   974
notation mset_le  (infix "\<subseteq>#" 50)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   975
notation mset_less  (infix "\<subset>#" 50)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   976
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   977
lemma mset_le_refl[simp]: "A \<le># A"
26178
nipkow
parents: 26176
diff changeset
   978
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   979
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   980
lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
26178
nipkow
parents: 26176
diff changeset
   981
unfolding mset_le_def by (fast intro: order_trans)
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   982
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   983
lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
26178
nipkow
parents: 26176
diff changeset
   984
apply (unfold mset_le_def)
nipkow
parents: 26176
diff changeset
   985
apply (rule multiset_eq_conv_count_eq [THEN iffD2])
nipkow
parents: 26176
diff changeset
   986
apply (blast intro: order_antisym)
nipkow
parents: 26176
diff changeset
   987
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   988
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   989
lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
26178
nipkow
parents: 26176
diff changeset
   990
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow
parents: 26176
diff changeset
   991
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
nipkow
parents: 26176
diff changeset
   992
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   993
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   994
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
26178
nipkow
parents: 26176
diff changeset
   995
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   996
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   997
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
26178
nipkow
parents: 26176
diff changeset
   998
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   999
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1000
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
26178
nipkow
parents: 26176
diff changeset
  1001
apply (unfold mset_le_def)
nipkow
parents: 26176
diff changeset
  1002
apply auto
nipkow
parents: 26176
diff changeset
  1003
apply (erule_tac x = a in allE)+
nipkow
parents: 26176
diff changeset
  1004
apply auto
nipkow
parents: 26176
diff changeset
  1005
done
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1006
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1007
lemma mset_le_add_left[simp]: "A \<le># A + B"
26178
nipkow
parents: 26176
diff changeset
  1008
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1009
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1010
lemma mset_le_add_right[simp]: "B \<le># A + B"
26178
nipkow
parents: 26176
diff changeset
  1011
unfolding mset_le_def by auto
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1012
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1013
lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
26178
nipkow
parents: 26176
diff changeset
  1014
by (simp add: mset_le_def)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1015
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1016
lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
26178
nipkow
parents: 26176
diff changeset
  1017
by (simp add: multiset_eq_conv_count_eq mset_le_def)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1018
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1019
lemma mset_le_multiset_union_diff_commute:
26178
nipkow
parents: 26176
diff changeset
  1020
assumes "B \<le># A"
nipkow
parents: 26176
diff changeset
  1021
shows "A - B + C = A + C - B"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1022
proof -
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1023
  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1024
  from this obtain D where "A = B + D" ..
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1025
  then show ?thesis
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1026
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1027
    apply (subst union_commute)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1028
    apply (subst multiset_diff_union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1029
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1030
    apply (simp add: diff_cancel)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1031
    apply (subst union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1032
    apply (subst union_commute[of "B" _])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1033
    apply (subst multiset_diff_union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1034
    apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1035
    apply (simp add: diff_cancel)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1036
    done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1037
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1038
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1039
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
26178
nipkow
parents: 26176
diff changeset
  1040
apply (induct xs)
nipkow
parents: 26176
diff changeset
  1041
 apply auto
nipkow
parents: 26176
diff changeset
  1042
apply (rule mset_le_trans)
nipkow
parents: 26176
diff changeset
  1043
 apply auto
nipkow
parents: 26176
diff changeset
  1044
done
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1045
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1046
lemma multiset_of_update:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1047
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1048
proof (induct ls arbitrary: i)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1049
  case Nil then show ?case by simp
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1050
next
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1051
  case (Cons x xs)
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1052
  show ?case
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1053
  proof (cases i)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1054
    case 0 then show ?thesis by simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1055
  next
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1056
    case (Suc i')
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1057
    with Cons show ?thesis
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1058
      apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1059
      apply (subst union_assoc)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1060
      apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1061
      apply (subst union_assoc [symmetric])
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1062
      apply simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1063
      apply (rule mset_le_multiset_union_diff_commute)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1064
      apply (simp add: mset_le_single nth_mem_multiset_of)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1065
      done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1066
  qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1067
qed
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1068
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1069
lemma multiset_of_swap:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1070
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1071
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
26178
nipkow
parents: 26176
diff changeset
  1072
apply (case_tac "i = j")
nipkow
parents: 26176
diff changeset
  1073
 apply simp
nipkow
parents: 26176
diff changeset
  1074
apply (simp add: multiset_of_update)
nipkow
parents: 26176
diff changeset
  1075
apply (subst elem_imp_eq_diff_union[symmetric])
nipkow
parents: 26176
diff changeset
  1076
 apply (simp add: nth_mem_multiset_of)
nipkow
parents: 26176
diff changeset
  1077
apply simp
nipkow
parents: 26176
diff changeset
  1078
done
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
  1079
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1080
interpretation mset_order!: order "op \<le>#" "op <#"
27682
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
  1081
proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
  1082
  mset_le_trans simp: mset_less_def)
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1083
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1084
interpretation mset_order_cancel_semigroup!:
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1085
  pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
27682
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
  1086
proof qed (erule mset_le_mono_add [OF mset_le_refl])
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
  1087
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1088
interpretation mset_order_semigroup_cancel!:
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1089
  pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
27682
25aceefd4786 added class preorder
haftmann
parents: 27611
diff changeset
  1090
proof qed simp
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
  1091
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1092
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1093
lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
26178
nipkow
parents: 26176
diff changeset
  1094
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1095
apply (erule_tac x=x in allE)
nipkow
parents: 26176
diff changeset
  1096
apply auto
nipkow
parents: 26176
diff changeset
  1097
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1098
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1099
lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
26178
nipkow
parents: 26176
diff changeset
  1100
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1101
apply (erule_tac x = x in allE)
nipkow
parents: 26176
diff changeset
  1102
apply auto
nipkow
parents: 26176
diff changeset
  1103
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1104
  
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1105
lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
26178
nipkow
parents: 26176
diff changeset
  1106
apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1107
 apply (simp add: mset_lessD)
nipkow
parents: 26176
diff changeset
  1108
apply (clarsimp simp: mset_le_def mset_less_def)
nipkow
parents: 26176
diff changeset
  1109
apply safe
nipkow
parents: 26176
diff changeset
  1110
 apply (erule_tac x = a in allE)
nipkow
parents: 26176
diff changeset
  1111
 apply (auto split: split_if_asm)
nipkow
parents: 26176
diff changeset
  1112
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1113
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1114
lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
26178
nipkow
parents: 26176
diff changeset
  1115
apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1116
 apply (simp add: mset_leD)
nipkow
parents: 26176
diff changeset
  1117
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
nipkow
parents: 26176
diff changeset
  1118
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1119
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1120
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" 
26178
nipkow
parents: 26176
diff changeset
  1121
by (induct A) (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1122
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1123
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
  1124
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1125
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1126
lemma multi_psub_self[simp]: "A \<subset># A = False"
26178
nipkow
parents: 26176
diff changeset
  1127
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1128
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1129
lemma mset_less_add_bothsides:
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1130
  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
26178
nipkow
parents: 26176
diff changeset
  1131
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1132
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1133
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
26178
nipkow
parents: 26176
diff changeset
  1134
by (auto simp: mset_le_def mset_less_def)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1135
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1136
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1137
proof (induct A arbitrary: B)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1138
  case (empty M)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1139
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1140
  then obtain M' x where "M = M' + {#x#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1141
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1142
  then show ?case by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1143
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1144
  case (add S x T)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1145
  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1146
  have SxsubT: "S + {#x#} \<subset># T" by fact
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1147
  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1148
  then obtain T' where T: "T = T' + {#x#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1149
    by (blast dest: multi_member_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1150
  then have "S \<subset># T'" using SxsubT 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1151
    by (blast intro: mset_less_add_bothsides)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1152
  then have "size S < size T'" using IH by simp
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1153
  then show ?case using T by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1154
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1155
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
  1156
lemmas mset_less_trans = mset_order.less_trans
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1157
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1158
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
26178
nipkow
parents: 26176
diff changeset
  1159
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1160
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1161
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1162
subsection {* Strong induction and subset induction for multisets *}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1163
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1164
text {* Well-foundedness of proper subset operator: *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1165
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1166
text {* proper multiset subset *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1167
definition
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1168
  mset_less_rel :: "('a multiset * 'a multiset) set" where
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1169
  "mset_less_rel = {(A,B). A \<subset># B}"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1170
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1171
lemma multiset_add_sub_el_shuffle: 
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1172
  assumes "c \<in># B" and "b \<noteq> c" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1173
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1174
proof -
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1175
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1176
    by (blast dest: multi_member_split)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1177
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1178
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1179
    by (simp add: union_ac)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1180
  then show ?thesis using B by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1181
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1182
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1183
lemma wf_mset_less_rel: "wf mset_less_rel"
26178
nipkow
parents: 26176
diff changeset
  1184
apply (unfold mset_less_rel_def)
nipkow
parents: 26176
diff changeset
  1185
apply (rule wf_measure [THEN wf_subset, where f1=size])
nipkow
parents: 26176
diff changeset
  1186
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
nipkow
parents: 26176
diff changeset
  1187
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1188
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1189
text {* The induction rules: *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1190
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1191
lemma full_multiset_induct [case_names less]:
26178
nipkow
parents: 26176
diff changeset
  1192
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
nipkow
parents: 26176
diff changeset
  1193
shows "P B"
nipkow
parents: 26176
diff changeset
  1194
apply (rule wf_mset_less_rel [THEN wf_induct])
nipkow
parents: 26176
diff changeset
  1195
apply (rule ih, auto simp: mset_less_rel_def)
nipkow
parents: 26176
diff changeset
  1196
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1197
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1198
lemma multi_subset_induct [consumes 2, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
  1199
assumes "F \<subseteq># A"
nipkow
parents: 26176
diff changeset
  1200
  and empty: "P {#}"
nipkow
parents: 26176
diff changeset
  1201
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
nipkow
parents: 26176
diff changeset
  1202
shows "P F"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1203
proof -
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1204
  from `F \<subseteq># A`
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1205
  show ?thesis
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1206
  proof (induct F)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1207
    show "P {#}" by fact
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1208
  next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1209
    fix x F
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1210
    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1211
    show "P (F + {#x#})"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1212
    proof (rule insert)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1213
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1214
      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1215
      with P show "P F" .
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1216
    qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1217
  qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1218
qed 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1219
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
  1220
text{* A consequence: Extensionality. *}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1221
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1222
lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
26178
nipkow
parents: 26176
diff changeset
  1223
apply (rule iffI)
nipkow
parents: 26176
diff changeset
  1224
 prefer 2
nipkow
parents: 26176
diff changeset
  1225
 apply clarsimp 
nipkow
parents: 26176
diff changeset
  1226
apply (induct A arbitrary: B rule: full_multiset_induct)
nipkow
parents: 26176
diff changeset
  1227
apply (rename_tac C)
nipkow
parents: 26176
diff changeset
  1228
apply (case_tac B rule: multiset_cases)
nipkow
parents: 26176
diff changeset
  1229
 apply (simp add: empty_multiset_count)
nipkow
parents: 26176
diff changeset
  1230
apply simp
nipkow
parents: 26176
diff changeset
  1231
apply (case_tac "x \<in># C")
nipkow
parents: 26176
diff changeset
  1232
 apply (force dest: multi_member_split)
nipkow
parents: 26176
diff changeset
  1233
apply (erule_tac x = x in allE)
nipkow
parents: 26176
diff changeset
  1234
apply simp
nipkow
parents: 26176
diff changeset
  1235
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1236
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1237
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1238
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1239
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1240
subsection {* The fold combinator *}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1241
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1242
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1243
  The intended behaviour is
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1244
  @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1245
  if @{text f} is associative-commutative. 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1246
*}
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1247
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1248
text {*
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1249
  The graph of @{text "fold_mset"}, @{text "z"}: the start element,
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1250
  @{text "f"}: folding function, @{text "A"}: the multiset, @{text
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1251
  "y"}: the result.
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1252
*}
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1253
inductive 
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1254
  fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1255
  for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1256
  and z :: 'b
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1257
where
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1258
  emptyI [intro]:  "fold_msetG f z {#} z"
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1259
| insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1260
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1261
inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1262
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1263
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1264
definition
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1265
  fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1266
  "fold_mset f z A = (THE x. fold_msetG f z A x)"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1267
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1268
lemma Diff1_fold_msetG:
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1269
  "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
26178
nipkow
parents: 26176
diff changeset
  1270
apply (frule_tac x = x in fold_msetG.insertI)
nipkow
parents: 26176
diff changeset
  1271
apply auto
nipkow
parents: 26176
diff changeset
  1272
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1273
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1274
lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
26178
nipkow
parents: 26176
diff changeset
  1275
apply (induct A)
nipkow
parents: 26176
diff changeset
  1276
 apply blast
nipkow
parents: 26176
diff changeset
  1277
apply clarsimp
nipkow
parents: 26176
diff changeset
  1278
apply (drule_tac x = x in fold_msetG.insertI)
nipkow
parents: 26176
diff changeset
  1279
apply auto
nipkow
parents: 26176
diff changeset
  1280
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1281
25759
6326138c1bd7 renamed foldM to fold_mset on general request
kleing
parents: 25623
diff changeset
  1282
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
26178
nipkow
parents: 26176
diff changeset
  1283
unfolding fold_mset_def by blast
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1284
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1285
locale left_commutative = 
26178
nipkow
parents: 26176
diff changeset
  1286
fixes f :: "'a => 'b => 'b"
nipkow
parents: 26176
diff changeset
  1287
assumes left_commute: "f x (f y z) = f y (f x z)"
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1288
begin
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1289
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1290
lemma fold_msetG_determ:
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
  1291
  "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1292
proof (induct arbitrary: x y z rule: full_multiset_induct)
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
  1293
  case (less M x\<^isub>1 x\<^isub>2 Z)