src/HOL/Library/Multiset.thy
author wenzelm
Wed, 14 Mar 2012 15:59:39 +0100
changeset 46921 aa862ff8a8a9
parent 46756 faf62905cd53
child 47143 212f7a975d49
permissions -rw-r--r--
some proof indentation;
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
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
     5
header {* (Finite) 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
46237
99c80c2f841a renamed theory AList to DAList
bulwahn
parents: 46168
diff changeset
     8
imports Main DAList
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
45694
4a8743618257 prefer typedef without extra definition and alternative name;
wenzelm
parents: 45608
diff changeset
    13
definition "multiset = {f :: 'a => nat. finite {x. f x > 0}}"
4a8743618257 prefer typedef without extra definition and alternative name;
wenzelm
parents: 45608
diff changeset
    14
4a8743618257 prefer typedef without extra definition and alternative name;
wenzelm
parents: 45608
diff changeset
    15
typedef (open) 'a multiset = "multiset :: ('a => nat) set"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    16
  morphisms count Abs_multiset
45694
4a8743618257 prefer typedef without extra definition and alternative name;
wenzelm
parents: 45608
diff changeset
    17
  unfolding multiset_def
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    18
proof
45694
4a8743618257 prefer typedef without extra definition and alternative name;
wenzelm
parents: 45608
diff changeset
    19
  show "(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}" by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    20
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    21
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    22
lemmas multiset_typedef = Abs_multiset_inverse count_inverse count
19086
1b3780be6cc2 new-style definitions/abbreviations;
wenzelm
parents: 18730
diff changeset
    23
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
    24
abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
    25
  "a :# M == 0 < count M a"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
    26
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    27
notation (xsymbols)
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
    28
  Melem (infix "\<in>#" 50)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    29
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
    30
lemma multiset_eq_iff:
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    31
  "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
    32
  by (simp only: count_inject [symmetric] fun_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    33
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
    34
lemma multiset_eqI:
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    35
  "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
    36
  using multiset_eq_iff by auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    37
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    38
text {*
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    39
 \medskip Preservation of the representing set @{term multiset}.
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    40
*}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    41
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    42
lemma const0_in_multiset:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    43
  "(\<lambda>a. 0) \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    44
  by (simp add: multiset_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    45
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    46
lemma only1_in_multiset:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    47
  "(\<lambda>b. if b = a then n else 0) \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    48
  by (simp add: multiset_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    49
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    50
lemma union_preserves_multiset:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    51
  "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    52
  by (simp add: multiset_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    53
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    54
lemma diff_preserves_multiset:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    55
  assumes "M \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    56
  shows "(\<lambda>a. M a - N a) \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    57
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    58
  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    59
    by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    60
  with assms show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    61
    by (auto simp add: multiset_def intro: finite_subset)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    62
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    63
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
    64
lemma filter_preserves_multiset:
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    65
  assumes "M \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    66
  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    67
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    68
  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    69
    by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    70
  with assms show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    71
    by (auto simp add: multiset_def intro: finite_subset)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    72
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    73
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    74
lemmas in_multiset = const0_in_multiset only1_in_multiset
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
    75
  union_preserves_multiset diff_preserves_multiset filter_preserves_multiset
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    76
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    77
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    78
subsection {* Representing multisets *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    79
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    80
text {* Multiset enumeration *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    81
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    82
instantiation multiset :: (type) "{zero, plus}"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    83
begin
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    84
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    85
definition Mempty_def:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    86
  "0 = Abs_multiset (\<lambda>a. 0)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    87
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    88
abbreviation Mempty :: "'a multiset" ("{#}") where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    89
  "Mempty \<equiv> 0"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    90
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    91
definition union_def:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    92
  "M + N = Abs_multiset (\<lambda>a. count M a + count N a)"
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    93
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    94
instance ..
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    95
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25507
diff changeset
    96
end
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
    97
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    98
definition single :: "'a => 'a multiset" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
    99
  "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   100
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   101
syntax
26176
038baad81209 tuned syntax declaration;
wenzelm
parents: 26145
diff changeset
   102
  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
25507
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
   103
translations
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
   104
  "{#x, xs#}" == "{#x#} + {#xs#}"
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
   105
  "{#x#}" == "CONST single x"
d13468d40131 added {#.,.,...#}
nipkow
parents: 25208
diff changeset
   106
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   107
lemma count_empty [simp]: "count {#} a = 0"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   108
  by (simp add: Mempty_def in_multiset multiset_typedef)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   109
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   110
lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   111
  by (simp add: single_def in_multiset multiset_typedef)
29901
f4b3f8fbf599 finiteness lemmas
nipkow
parents: 29509
diff changeset
   112
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   113
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   114
subsection {* Basic operations *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   115
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   116
subsubsection {* Union *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   117
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   118
lemma count_union [simp]: "count (M + N) a = count M a + count N a"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   119
  by (simp add: union_def in_multiset multiset_typedef)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   120
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   121
instance multiset :: (type) cancel_comm_monoid_add
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   122
  by default (simp_all add: multiset_eq_iff)
10277
081c8641aa11 improved typedef;
wenzelm
parents: 10249
diff changeset
   123
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   124
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   125
subsubsection {* Difference *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   126
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   127
instantiation multiset :: (type) minus
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   128
begin
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   129
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   130
definition diff_def:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   131
  "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   132
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   133
instance ..
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   134
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   135
end
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   136
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   137
lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   138
  by (simp add: diff_def in_multiset multiset_typedef)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   139
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   140
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   141
by(simp add: multiset_eq_iff)
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   142
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   143
lemma diff_cancel[simp]: "A - A = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   144
by (rule multiset_eqI) simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   145
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   146
lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   147
by(simp add: multiset_eq_iff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   148
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   149
lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   150
by(simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   151
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   152
lemma insert_DiffM:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   153
  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   154
  by (clarsimp simp: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   155
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   156
lemma insert_DiffM2 [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   157
  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   158
  by (clarsimp simp: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   159
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   160
lemma diff_right_commute:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   161
  "(M::'a multiset) - N - Q = M - Q - N"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   162
  by (auto simp add: multiset_eq_iff)
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   163
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   164
lemma diff_add:
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   165
  "(M::'a multiset) - (N + Q) = M - N - Q"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   166
by (simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   167
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   168
lemma diff_union_swap:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   169
  "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   170
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   171
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   172
lemma diff_union_single_conv:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   173
  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   174
  by (simp add: multiset_eq_iff)
26143
314c0bcb7df7 Added useful general lemmas from the work with the HeapMonad
bulwahn
parents: 26033
diff changeset
   175
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   176
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   177
subsubsection {* Equality of multisets *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   178
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   179
lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   180
  by (simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   181
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   182
lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   183
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   184
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   185
lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   186
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   187
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   188
lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   189
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   190
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   191
lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   192
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   193
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   194
lemma diff_single_trivial:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   195
  "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   196
  by (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   197
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   198
lemma diff_single_eq_union:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   199
  "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   200
  by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   201
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   202
lemma union_single_eq_diff:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   203
  "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   204
  by (auto dest: sym)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   205
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   206
lemma union_single_eq_member:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   207
  "M + {#x#} = N \<Longrightarrow> x \<in># N"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   208
  by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   209
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   210
lemma union_is_single:
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   211
  "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   212
proof
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   213
  assume ?rhs then show ?lhs by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   214
next
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   215
  assume ?lhs then show ?rhs
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   216
    by (simp add: multiset_eq_iff split:if_splits) (metis add_is_1)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   217
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   218
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   219
lemma single_is_union:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   220
  "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   221
  by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   222
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   223
lemma add_eq_conv_diff:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   224
  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44339
diff changeset
   225
(* shorter: by (simp add: multiset_eq_iff) fastforce *)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   226
proof
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   227
  assume ?rhs then show ?lhs
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   228
  by (auto simp add: add_assoc add_commute [of "{#b#}"])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   229
    (drule sym, simp add: add_assoc [symmetric])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   230
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   231
  assume ?lhs
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   232
  show ?rhs
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   233
  proof (cases "a = b")
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   234
    case True with `?lhs` show ?thesis by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   235
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   236
    case False
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   237
    from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   238
    with False have "a \<in># N" by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   239
    moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   240
    moreover note False
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   241
    ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   242
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   243
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   244
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   245
lemma insert_noteq_member: 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   246
  assumes BC: "B + {#b#} = C + {#c#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   247
   and bnotc: "b \<noteq> c"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   248
  shows "c \<in># B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   249
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   250
  have "c \<in># C + {#c#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   251
  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   252
  then have "c \<in># B + {#b#}" using BC by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   253
  then show "c \<in># B" using nc by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   254
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   255
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   256
lemma add_eq_conv_ex:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   257
  "(M + {#a#} = N + {#b#}) =
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   258
    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   259
  by (auto simp add: add_eq_conv_diff)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   260
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   261
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   262
subsubsection {* Pointwise ordering induced by count *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   263
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   264
instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   265
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   266
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   267
definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   268
  mset_le_def: "A \<le> B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   269
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   270
definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   271
  mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   272
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   273
instance
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   274
  by default (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   275
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   276
end
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   277
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   278
lemma mset_less_eqI:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   279
  "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   280
  by (simp add: mset_le_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   281
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   282
lemma mset_le_exists_conv:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   283
  "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   284
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   285
apply (auto intro: multiset_eq_iff [THEN iffD2])
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   286
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   287
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   288
lemma mset_le_mono_add_right_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   289
  "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   290
  by (fact add_le_cancel_right)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   291
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   292
lemma mset_le_mono_add_left_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   293
  "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   294
  by (fact add_le_cancel_left)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   295
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   296
lemma mset_le_mono_add:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   297
  "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   298
  by (fact add_mono)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   299
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   300
lemma mset_le_add_left [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   301
  "(A::'a multiset) \<le> A + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   302
  unfolding mset_le_def by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   303
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   304
lemma mset_le_add_right [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   305
  "B \<le> (A::'a multiset) + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   306
  unfolding mset_le_def by auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   307
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   308
lemma mset_le_single:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   309
  "a :# B \<Longrightarrow> {#a#} \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   310
  by (simp add: mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   311
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   312
lemma multiset_diff_union_assoc:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   313
  "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   314
  by (simp add: multiset_eq_iff mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   315
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   316
lemma mset_le_multiset_union_diff_commute:
36867
6c28c702ed22 simplified proof
nipkow
parents: 36635
diff changeset
   317
  "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   318
by (simp add: multiset_eq_iff mset_le_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   319
39301
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   320
lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   321
by(simp add: mset_le_def)
e1bd8a54c40f added and renamed lemmas
nipkow
parents: 39198
diff changeset
   322
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   323
lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   324
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   325
apply (erule_tac x=x in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   326
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   327
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   328
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   329
lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   330
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   331
apply (erule_tac x = x in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   332
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   333
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   334
  
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   335
lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   336
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   337
 apply (simp add: mset_lessD)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   338
apply (clarsimp simp: mset_le_def mset_less_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   339
apply safe
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   340
 apply (erule_tac x = a in allE)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   341
 apply (auto split: split_if_asm)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   342
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   343
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   344
lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   345
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   346
 apply (simp add: mset_leD)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   347
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   348
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   349
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   350
lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   351
  by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   352
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   353
lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   354
  by (auto simp: mset_le_def mset_less_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   355
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   356
lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   357
  by simp
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   358
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   359
lemma mset_less_add_bothsides:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   360
  "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   361
  by (fact add_less_imp_less_right)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   362
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   363
lemma mset_less_empty_nonempty:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   364
  "{#} < S \<longleftrightarrow> S \<noteq> {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   365
  by (auto simp: mset_le_def mset_less_def)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   366
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   367
lemma mset_less_diff_self:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   368
  "c \<in># B \<Longrightarrow> B - {#c#} < B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   369
  by (auto simp: mset_le_def mset_less_def multiset_eq_iff)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   370
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   371
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   372
subsubsection {* Intersection *}
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   373
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   374
instantiation multiset :: (type) semilattice_inf
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   375
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   376
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   377
definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   378
  multiset_inter_def: "inf_multiset A B = A - (A - B)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   379
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   380
instance
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   381
proof -
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   382
  have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   383
  show "OFCLASS('a multiset, semilattice_inf_class)"
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   384
    by default (auto simp add: multiset_inter_def mset_le_def aux)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   385
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   386
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   387
end
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   388
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   389
abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   390
  "multiset_inter \<equiv> inf"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   391
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   392
lemma multiset_inter_count [simp]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   393
  "count (A #\<inter> B) x = min (count A x) (count B x)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   394
  by (simp add: multiset_inter_def multiset_typedef)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   395
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   396
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   397
  by (rule multiset_eqI) auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   398
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   399
lemma multiset_union_diff_commute:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   400
  assumes "B #\<inter> C = {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   401
  shows "A + B - C = A - C + B"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   402
proof (rule multiset_eqI)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   403
  fix x
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   404
  from assms have "min (count B x) (count C x) = 0"
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   405
    by (auto simp add: multiset_eq_iff)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   406
  then have "count B x = 0 \<or> count C x = 0"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   407
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   408
  then show "count (A + B - C) x = count (A - C + B) x"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   409
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   410
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   411
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   412
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   413
subsubsection {* Filter (with comprehension syntax) *}
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   414
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   415
text {* Multiset comprehension *}
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   416
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   417
definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   418
  "filter P M = Abs_multiset (\<lambda>x. if P x then count M x else 0)"
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   419
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   420
hide_const (open) filter
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   421
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   422
lemma count_filter [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   423
  "count (Multiset.filter P M) a = (if P a then count M a else 0)"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   424
  by (simp add: filter_def in_multiset multiset_typedef)
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   425
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   426
lemma filter_empty [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   427
  "Multiset.filter P {#} = {#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   428
  by (rule multiset_eqI) simp
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   429
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   430
lemma filter_single [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   431
  "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   432
  by (rule multiset_eqI) simp
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   433
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   434
lemma filter_union [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   435
  "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   436
  by (rule multiset_eqI) simp
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   437
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   438
lemma filter_diff [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   439
  "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   440
  by (rule multiset_eqI) simp
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   441
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   442
lemma filter_inter [simp]:
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   443
  "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   444
  by (rule multiset_eqI) simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   445
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   446
syntax
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   447
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   448
syntax (xsymbol)
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   449
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   450
translations
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   451
  "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   452
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   453
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   454
subsubsection {* Set of elements *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   455
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   456
definition set_of :: "'a multiset => 'a set" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   457
  "set_of M = {x. x :# M}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   458
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   459
lemma set_of_empty [simp]: "set_of {#} = {}"
26178
nipkow
parents: 26176
diff changeset
   460
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   461
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   462
lemma set_of_single [simp]: "set_of {#b#} = {b}"
26178
nipkow
parents: 26176
diff changeset
   463
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   464
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   465
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
26178
nipkow
parents: 26176
diff changeset
   466
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   467
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   468
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   469
by (auto simp add: set_of_def multiset_eq_iff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   470
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   471
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
26178
nipkow
parents: 26176
diff changeset
   472
by (auto simp add: set_of_def)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   473
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   474
lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
26178
nipkow
parents: 26176
diff changeset
   475
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   476
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   477
lemma finite_set_of [iff]: "finite (set_of M)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   478
  using count [of M] by (simp add: multiset_def set_of_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   479
46756
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46730
diff changeset
   480
lemma finite_Collect_mem [iff]: "finite {x. x :# M}"
faf62905cd53 adding finiteness of intervals on integer sets; adding another finiteness theorem for multisets
bulwahn
parents: 46730
diff changeset
   481
  unfolding set_of_def[symmetric] by simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   482
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   483
subsubsection {* Size *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   484
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   485
instantiation multiset :: (type) size
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   486
begin
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   487
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   488
definition size_def:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   489
  "size M = setsum (count M) (set_of M)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   490
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   491
instance ..
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   492
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   493
end
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   494
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   495
lemma size_empty [simp]: "size {#} = 0"
26178
nipkow
parents: 26176
diff changeset
   496
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   497
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   498
lemma size_single [simp]: "size {#b#} = 1"
26178
nipkow
parents: 26176
diff changeset
   499
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   500
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   501
lemma setsum_count_Int:
26178
nipkow
parents: 26176
diff changeset
   502
  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow
parents: 26176
diff changeset
   503
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   504
 apply simp
nipkow
parents: 26176
diff changeset
   505
apply (simp add: Int_insert_left set_of_def)
nipkow
parents: 26176
diff changeset
   506
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   507
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   508
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
26178
nipkow
parents: 26176
diff changeset
   509
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   510
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow
parents: 26176
diff changeset
   511
 prefer 2
nipkow
parents: 26176
diff changeset
   512
 apply (rule ext, simp)
nipkow
parents: 26176
diff changeset
   513
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow
parents: 26176
diff changeset
   514
apply (subst Int_commute)
nipkow
parents: 26176
diff changeset
   515
apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow
parents: 26176
diff changeset
   516
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   517
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   518
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   519
by (auto simp add: size_def multiset_eq_iff)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   520
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   521
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
26178
nipkow
parents: 26176
diff changeset
   522
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
   523
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   524
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
26178
nipkow
parents: 26176
diff changeset
   525
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   526
apply (drule setsum_SucD)
nipkow
parents: 26176
diff changeset
   527
apply auto
nipkow
parents: 26176
diff changeset
   528
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   529
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   530
lemma size_eq_Suc_imp_eq_union:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   531
  assumes "size M = Suc n"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   532
  shows "\<exists>a N. M = N + {#a#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   533
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   534
  from assms obtain a where "a \<in># M"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   535
    by (erule size_eq_Suc_imp_elem [THEN exE])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   536
  then have "M = M - {#a#} + {#a#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   537
  then show ?thesis by blast
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   538
qed
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   539
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   540
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   541
subsection {* Induction and case splits *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   542
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   543
lemma setsum_decr:
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   544
  "finite F ==> (0::nat) < f a ==>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   545
    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
   546
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   547
 apply auto
nipkow
parents: 26176
diff changeset
   548
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow
parents: 26176
diff changeset
   549
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   550
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   551
lemma rep_multiset_induct_aux:
26178
nipkow
parents: 26176
diff changeset
   552
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow
parents: 26176
diff changeset
   553
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow
parents: 26176
diff changeset
   554
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow
parents: 26176
diff changeset
   555
apply (unfold multiset_def)
nipkow
parents: 26176
diff changeset
   556
apply (induct_tac n, simp, clarify)
nipkow
parents: 26176
diff changeset
   557
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow
parents: 26176
diff changeset
   558
  apply simp
nipkow
parents: 26176
diff changeset
   559
  apply (rule 1)
nipkow
parents: 26176
diff changeset
   560
 apply (rule ext, force, clarify)
nipkow
parents: 26176
diff changeset
   561
apply (frule setsum_SucD, clarify)
nipkow
parents: 26176
diff changeset
   562
apply (rename_tac a)
nipkow
parents: 26176
diff changeset
   563
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow
parents: 26176
diff changeset
   564
 prefer 2
nipkow
parents: 26176
diff changeset
   565
 apply (rule finite_subset)
nipkow
parents: 26176
diff changeset
   566
  prefer 2
nipkow
parents: 26176
diff changeset
   567
  apply assumption
nipkow
parents: 26176
diff changeset
   568
 apply simp
nipkow
parents: 26176
diff changeset
   569
 apply blast
nipkow
parents: 26176
diff changeset
   570
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow
parents: 26176
diff changeset
   571
 prefer 2
nipkow
parents: 26176
diff changeset
   572
 apply (rule ext)
nipkow
parents: 26176
diff changeset
   573
 apply (simp (no_asm_simp))
nipkow
parents: 26176
diff changeset
   574
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow
parents: 26176
diff changeset
   575
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow
parents: 26176
diff changeset
   576
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow
parents: 26176
diff changeset
   577
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow
parents: 26176
diff changeset
   578
 prefer 2
nipkow
parents: 26176
diff changeset
   579
 apply blast
nipkow
parents: 26176
diff changeset
   580
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow
parents: 26176
diff changeset
   581
 prefer 2
nipkow
parents: 26176
diff changeset
   582
 apply blast
nipkow
parents: 26176
diff changeset
   583
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow
parents: 26176
diff changeset
   584
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   585
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   586
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   587
  "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
   588
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
26178
nipkow
parents: 26176
diff changeset
   589
using rep_multiset_induct_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   590
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   591
theorem multiset_induct [case_names empty add, induct type: multiset]:
26178
nipkow
parents: 26176
diff changeset
   592
assumes empty: "P {#}"
nipkow
parents: 26176
diff changeset
   593
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow
parents: 26176
diff changeset
   594
shows "P M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   595
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   596
  note defns = union_def single_def Mempty_def
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   597
  note add' = add [unfolded defns, simplified]
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   598
  have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) =
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   599
    (\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset) 
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   600
  show ?thesis
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   601
    apply (rule count_inverse [THEN subst])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   602
    apply (rule count [THEN rep_multiset_induct])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   603
     apply (rule empty [unfolded defns])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   604
    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
   605
     prefer 2
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   606
     apply (simp add: fun_eq_iff)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   607
    apply (erule ssubst)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   608
    apply (erule Abs_multiset_inverse [THEN subst])
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   609
    apply (drule add')
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   610
    apply (simp add: aux)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   611
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   612
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   613
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   614
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   615
by (induct M) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   616
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   617
lemma multiset_cases [cases type, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
   618
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   619
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   620
shows "P"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   621
proof (cases "M = {#}")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   622
  assume "M = {#}" then show ?thesis using em by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   623
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   624
  assume "M \<noteq> {#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   625
  then obtain M' m where "M = M' + {#m#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   626
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   627
  then show ?thesis using add by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   628
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   629
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   630
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
   631
apply (cases M)
nipkow
parents: 26176
diff changeset
   632
 apply simp
nipkow
parents: 26176
diff changeset
   633
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow
parents: 26176
diff changeset
   634
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   635
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   636
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   637
by (cases "B = {#}") (auto dest: multi_member_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   638
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   639
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   640
apply (subst multiset_eq_iff)
26178
nipkow
parents: 26176
diff changeset
   641
apply auto
nipkow
parents: 26176
diff changeset
   642
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   643
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   644
lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   645
proof (induct A arbitrary: B)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   646
  case (empty M)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   647
  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   648
  then obtain M' x where "M = M' + {#x#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   649
    by (blast dest: multi_nonempty_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   650
  then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   651
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   652
  case (add S x T)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   653
  have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   654
  have SxsubT: "S + {#x#} < T" by fact
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   655
  then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   656
  then obtain T' where T: "T = T' + {#x#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   657
    by (blast dest: multi_member_split)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   658
  then have "S < T'" using SxsubT 
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   659
    by (blast intro: mset_less_add_bothsides)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   660
  then have "size S < size T'" using IH by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   661
  then show ?case using T by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   662
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   663
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   664
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   665
subsubsection {* Strong induction and subset induction for multisets *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   666
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   667
text {* Well-foundedness of proper subset operator: *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   668
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   669
text {* proper multiset subset *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   670
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   671
definition
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   672
  mset_less_rel :: "('a multiset * 'a multiset) set" where
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   673
  "mset_less_rel = {(A,B). A < B}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   674
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   675
lemma multiset_add_sub_el_shuffle: 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   676
  assumes "c \<in># B" and "b \<noteq> c" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   677
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   678
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   679
  from `c \<in># B` obtain A where B: "B = A + {#c#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   680
    by (blast dest: multi_member_split)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   681
  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   682
  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   683
    by (simp add: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   684
  then show ?thesis using B by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   685
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   686
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   687
lemma wf_mset_less_rel: "wf mset_less_rel"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   688
apply (unfold mset_less_rel_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   689
apply (rule wf_measure [THEN wf_subset, where f1=size])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   690
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   691
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   692
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   693
text {* The induction rules: *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   694
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   695
lemma full_multiset_induct [case_names less]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   696
assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   697
shows "P B"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   698
apply (rule wf_mset_less_rel [THEN wf_induct])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   699
apply (rule ih, auto simp: mset_less_rel_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   700
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   701
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   702
lemma multi_subset_induct [consumes 2, case_names empty add]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   703
assumes "F \<le> A"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   704
  and empty: "P {#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   705
  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   706
shows "P F"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   707
proof -
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   708
  from `F \<le> A`
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   709
  show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   710
  proof (induct F)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   711
    show "P {#}" by fact
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   712
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   713
    fix x F
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   714
    assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   715
    show "P (F + {#x#})"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   716
    proof (rule insert)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   717
      from i show "x \<in># A" by (auto dest: mset_le_insertD)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   718
      from i have "F \<le> A" by (auto dest: mset_le_insertD)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   719
      with P show "P F" .
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   720
    qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   721
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   722
qed
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   723
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   724
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   725
subsection {* Alternative representations *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   726
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   727
subsubsection {* Lists *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   728
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   729
primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   730
  "multiset_of [] = {#}" |
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   731
  "multiset_of (a # x) = multiset_of x + {# a #}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   732
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   733
lemma in_multiset_in_set:
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   734
  "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   735
  by (induct xs) simp_all
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   736
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   737
lemma count_multiset_of:
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   738
  "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   739
  by (induct xs) simp_all
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   740
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   741
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   742
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   743
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   744
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   745
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   746
40950
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   747
lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   748
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   749
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   750
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   751
by (induct xs) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   752
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   753
lemma multiset_of_append [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   754
  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   755
  by (induct xs arbitrary: ys) (auto simp: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   756
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   757
lemma multiset_of_filter:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   758
  "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   759
  by (induct xs) simp_all
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   760
40950
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   761
lemma multiset_of_rev [simp]:
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   762
  "multiset_of (rev xs) = multiset_of xs"
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   763
  by (induct xs) simp_all
a370b0fb6f09 lemma multiset_of_rev
haftmann
parents: 40606
diff changeset
   764
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   765
lemma surj_multiset_of: "surj multiset_of"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   766
apply (unfold surj_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   767
apply (rule allI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   768
apply (rule_tac M = y in multiset_induct)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   769
 apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   770
apply (rule_tac x = "x # xa" in exI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   771
apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   772
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   773
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   774
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   775
by (induct x) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   776
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   777
lemma distinct_count_atmost_1:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   778
  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   779
apply (induct x, simp, rule iffI, simp_all)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   780
apply (rule conjI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   781
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   782
apply (erule_tac x = a in allE, simp, clarify)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   783
apply (erule_tac x = aa in allE, simp)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   784
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   785
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   786
lemma multiset_of_eq_setD:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   787
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   788
by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   789
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   790
lemma set_eq_iff_multiset_of_eq_distinct:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   791
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   792
    (set x = set y) = (multiset_of x = multiset_of y)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   793
by (auto simp: multiset_eq_iff distinct_count_atmost_1)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   794
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   795
lemma set_eq_iff_multiset_of_remdups_eq:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   796
   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   797
apply (rule iffI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   798
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   799
apply (drule distinct_remdups [THEN distinct_remdups
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   800
      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   801
apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   802
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   803
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   804
lemma multiset_of_compl_union [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   805
  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   806
  by (induct xs) (auto simp: add_ac)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   807
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   808
lemma count_multiset_of_length_filter:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   809
  "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   810
  by (induct xs) auto
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   811
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   812
lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   813
apply (induct ls arbitrary: i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   814
 apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   815
apply (case_tac i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   816
 apply auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   817
done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   818
36903
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   819
lemma multiset_of_remove1[simp]:
489c1fbbb028 Multiset: renamed, added and tuned lemmas;
nipkow
parents: 36867
diff changeset
   820
  "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
   821
by (induct xs) (auto simp add: multiset_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   822
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   823
lemma multiset_of_eq_length:
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   824
  assumes "multiset_of xs = multiset_of ys"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   825
  shows "length xs = length ys"
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   826
using assms
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   827
proof (induct xs arbitrary: ys)
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   828
  case Nil then show ?case by simp
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   829
next
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   830
  case (Cons x xs)
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   831
  then have "x \<in># multiset_of ys" by (simp add: union_single_eq_member)
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   832
  then have "x \<in> set ys" by (simp add: in_multiset_in_set)
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   833
  from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)"
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   834
    by simp
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   835
  with Cons.hyps have "length xs = length (remove1 x ys)" .
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   836
  with `x \<in> set ys` show ?case
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   837
    by (auto simp add: length_remove1 dest: length_pos_if_in_set)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   838
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   839
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   840
lemma multiset_of_eq_length_filter:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   841
  assumes "multiset_of xs = multiset_of ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   842
  shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   843
proof (cases "z \<in># multiset_of xs")
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   844
  case False
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   845
  moreover have "\<not> z \<in># multiset_of ys" using assms False by simp
41069
6fabc0414055 name filter operation just filter (c.f. List.filter and list comprehension syntax)
haftmann
parents: 40968
diff changeset
   846
  ultimately show ?thesis by (simp add: count_multiset_of_length_filter)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   847
next
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   848
  case True
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   849
  moreover have "z \<in># multiset_of ys" using assms True by simp
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   850
  show ?thesis using assms
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   851
  proof (induct xs arbitrary: ys)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   852
    case Nil then show ?case by simp
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   853
  next
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   854
    case (Cons x xs)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   855
    from `multiset_of (x # xs) = multiset_of ys` [symmetric]
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   856
      have *: "multiset_of xs = multiset_of (remove1 x ys)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   857
      and "x \<in> set ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   858
      by (auto simp add: mem_set_multiset_eq)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   859
    from * have "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) (remove1 x ys))" by (rule Cons.hyps)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   860
    moreover from `x \<in> set ys` have "length (filter (\<lambda>y. x = y) ys) > 0" by (simp add: filter_empty_conv)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   861
    ultimately show ?case using `x \<in> set ys`
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   862
      by (simp add: filter_remove1) (auto simp add: length_remove1)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   863
  qed
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   864
qed
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   865
45989
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   866
lemma fold_multiset_equiv:
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   867
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   868
    and equiv: "multiset_of xs = multiset_of ys"
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   869
  shows "fold f xs = fold f ys"
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   870
using f equiv [symmetric]
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   871
proof (induct xs arbitrary: ys)
45989
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   872
  case Nil then show ?case by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   873
next
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   874
  case (Cons x xs)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   875
  then have *: "set ys = set (x # xs)" by (blast dest: multiset_of_eq_setD)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   876
  have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x" 
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   877
    by (rule Cons.prems(1)) (simp_all add: *)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   878
  moreover from * have "x \<in> set ys" by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   879
  ultimately have "fold f ys = fold f (remove1 x ys) \<circ> f x" by (fact fold_remove1_split)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   880
  moreover from Cons.prems have "fold f xs = fold f (remove1 x ys)" by (auto intro: Cons.hyps)
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   881
  ultimately show ?case by simp
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   882
qed
b39256df5f8a moved theorem requiring multisets from More_List to Multiset
haftmann
parents: 45866
diff changeset
   883
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   884
context linorder
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   885
begin
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   886
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 39533
diff changeset
   887
lemma multiset_of_insort [simp]:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   888
  "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   889
  by (induct xs) (simp_all add: ac_simps)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   890
 
40210
aee7ef725330 sorting: avoid _key suffix if lemma applies both to simple and generalized variant; generalized insort_insert to insort_insert_key; additional lemmas
haftmann
parents: 39533
diff changeset
   891
lemma multiset_of_sort [simp]:
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   892
  "multiset_of (sort_key k xs) = multiset_of xs"
37107
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   893
  by (induct xs) (simp_all add: ac_simps)
1535aa1c943a more lemmas
haftmann
parents: 37074
diff changeset
   894
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   895
text {*
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   896
  This lemma shows which properties suffice to show that a function
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   897
  @{text "f"} with @{text "f xs = ys"} behaves like sort.
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   898
*}
37074
322d065ebef7 localized properties_for_sort
haftmann
parents: 36903
diff changeset
   899
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   900
lemma properties_for_sort_key:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   901
  assumes "multiset_of ys = multiset_of xs"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   902
  and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   903
  and "sorted (map f ys)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   904
  shows "sort_key f xs = ys"
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   905
using assms
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
   906
proof (induct xs arbitrary: ys)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   907
  case Nil then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   908
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   909
  case (Cons x xs)
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   910
  from Cons.prems(2) have
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   911
    "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   912
    by (simp add: filter_remove1)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   913
  with Cons.prems have "sort_key f xs = remove1 x ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   914
    by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   915
  moreover from Cons.prems have "x \<in> set ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   916
    by (auto simp add: mem_set_multiset_eq intro!: ccontr)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   917
  ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   918
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   919
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   920
lemma properties_for_sort:
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   921
  assumes multiset: "multiset_of ys = multiset_of xs"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   922
  and "sorted ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   923
  shows "sort xs = ys"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   924
proof (rule properties_for_sort_key)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   925
  from multiset show "multiset_of ys = multiset_of xs" .
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   926
  from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   927
  from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   928
    by (rule multiset_of_eq_length_filter)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   929
  then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   930
    by simp
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   931
  then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   932
    by (simp add: replicate_length_filter)
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   933
qed
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
   934
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   935
lemma sort_key_by_quicksort:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   936
  "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   937
    @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   938
    @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   939
proof (rule properties_for_sort_key)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   940
  show "multiset_of ?rhs = multiset_of ?lhs"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   941
    by (rule multiset_eqI) (auto simp add: multiset_of_filter)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   942
next
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   943
  show "sorted (map f ?rhs)"
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   944
    by (auto simp add: sorted_append intro: sorted_map_same)
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   945
next
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   946
  fix l
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   947
  assume "l \<in> set ?rhs"
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   948
  let ?pivot = "f (xs ! (length xs div 2))"
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   949
  have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
   950
  have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   951
    unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   952
  with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   953
  have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   954
  then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   955
    [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   956
  note *** = this [of "op <"] this [of "op >"] this [of "op ="]
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
   957
  show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   958
  proof (cases "f l" ?pivot rule: linorder_cases)
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   959
    case less
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   960
    then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   961
    with less show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   962
      by (simp add: filter_sort [symmetric] ** ***)
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   963
  next
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
   964
    case equal then show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   965
      by (simp add: * less_le)
40305
41833242cc42 tuned lemma proposition of properties_for_sort_key
haftmann
parents: 40303
diff changeset
   966
  next
46730
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   967
    case greater
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   968
    then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
e3b99d0231bc tuned proofs;
wenzelm
parents: 46394
diff changeset
   969
    with greater show ?thesis
40346
58af2b8327b7 tuned proof
haftmann
parents: 40307
diff changeset
   970
      by (simp add: filter_sort [symmetric] ** ***)
40306
e4461b9854a5 tuned proof
haftmann
parents: 40305
diff changeset
   971
  qed
40303
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   972
qed
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   973
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   974
lemma sort_by_quicksort:
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   975
  "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   976
    @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   977
    @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   978
  using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp
2d507370e879 lemmas multiset_of_filter, sort_key_by_quicksort
haftmann
parents: 40250
diff changeset
   979
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   980
text {* A stable parametrized quicksort *}
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   981
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   982
definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   983
  "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   984
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   985
lemma part_code [code]:
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   986
  "part f pivot [] = ([], [], [])"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   987
  "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   988
     if x' < pivot then (x # lts, eqs, gts)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   989
     else if x' > pivot then (lts, eqs, x # gts)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   990
     else (lts, x # eqs, gts))"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   991
  by (auto simp add: part_def Let_def split_def)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   992
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   993
lemma sort_key_by_quicksort_code [code]:
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   994
  "sort_key f xs = (case xs of [] \<Rightarrow> []
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   995
    | [x] \<Rightarrow> xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   996
    | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   997
    | _ \<Rightarrow> (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   998
       in sort_key f lts @ eqs @ sort_key f gts))"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
   999
proof (cases xs)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1000
  case Nil then show ?thesis by simp
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1001
next
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1002
  case (Cons _ ys) note hyps = Cons show ?thesis
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1003
  proof (cases ys)
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1004
    case Nil with hyps show ?thesis by simp
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1005
  next
46921
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1006
    case (Cons _ zs) note hyps = hyps Cons show ?thesis
aa862ff8a8a9 some proof indentation;
wenzelm
parents: 46756
diff changeset
  1007
    proof (cases zs)
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1008
      case Nil with hyps show ?thesis by auto
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1009
    next
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1010
      case Cons 
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1011
      from sort_key_by_quicksort [of f xs]
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1012
      have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1013
        in sort_key f lts @ eqs @ sort_key f gts)"
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1014
      by (simp only: split_def Let_def part_def fst_conv snd_conv)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1015
      with hyps Cons show ?thesis by (simp only: list.cases)
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1016
    qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1017
  qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1018
qed
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1019
39533
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1020
end
91a0ff0ff237 generalized lemmas multiset_of_insort, multiset_of_sort, properties_for_sort for *_key variants
haftmann
parents: 39314
diff changeset
  1021
40347
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1022
hide_const (open) part
429bf4388b2f added code lemmas for stable parametrized quicksort
haftmann
parents: 40346
diff changeset
  1023
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
  1024
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
  1025
  by (induct xs) (auto intro: order_trans)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1026
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1027
lemma multiset_of_update:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1028
  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1029
proof (induct ls arbitrary: i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1030
  case Nil then show ?case by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1031
next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1032
  case (Cons x xs)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1033
  show ?case
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1034
  proof (cases i)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1035
    case 0 then show ?thesis by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1036
  next
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1037
    case (Suc i')
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1038
    with Cons show ?thesis
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1039
      apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1040
      apply (subst add_assoc)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1041
      apply (subst add_commute [of "{#v#}" "{#x#}"])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1042
      apply (subst add_assoc [symmetric])
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1043
      apply simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1044
      apply (rule mset_le_multiset_union_diff_commute)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1045
      apply (simp add: mset_le_single nth_mem_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1046
      done
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1047
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1048
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1049
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1050
lemma multiset_of_swap:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1051
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1052
    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1053
  by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1054
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1055
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1056
subsubsection {* Association lists -- including code generation *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1057
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1058
text {* Preliminaries *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1059
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1060
text {* Raw operations on lists *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1061
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1062
definition join_raw :: "('key \<Rightarrow> 'val \<times> 'val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1063
where
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1064
  "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (%v'. f k (v', v))) ys xs"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1065
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1066
lemma join_raw_Nil [simp]:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1067
  "join_raw f xs [] = xs"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1068
by (simp add: join_raw_def)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1069
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1070
lemma join_raw_Cons [simp]:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1071
  "join_raw f xs ((k, v) # ys) = map_default k v (%v'. f k (v', v)) (join_raw f xs ys)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1072
by (simp add: join_raw_def)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1073
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1074
lemma map_of_join_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1075
  assumes "distinct (map fst ys)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1076
  shows "map_of (join_raw f xs ys) x = (case map_of xs x of None => map_of ys x | Some v => (case map_of ys x of None => Some v | Some v' => Some (f x (v, v'))))"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1077
using assms
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1078
apply (induct ys)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1079
apply (auto simp add: map_of_map_default split: option.split)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1080
apply (metis map_of_eq_None_iff option.simps(2) weak_map_of_SomeI)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1081
by (metis Some_eq_map_of_iff map_of_eq_None_iff option.simps(2))
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1082
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1083
lemma distinct_join_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1084
  assumes "distinct (map fst xs)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1085
  shows "distinct (map fst (join_raw f xs ys))"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1086
using assms
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1087
proof (induct ys)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1088
  case (Cons y ys)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1089
  thus ?case by (cases y) (simp add: distinct_map_default)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1090
qed auto
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1091
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1092
definition
46238
9ace9e5b79be renaming theory AList_Impl back to AList (reverting 1fec5b365f9b; AList with distinct key invariant is called DAList)
bulwahn
parents: 46237
diff changeset
  1093
  "subtract_entries_raw xs ys = foldr (%(k, v). AList.map_entry k (%v'. v' - v)) ys xs"
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1094
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1095
lemma map_of_subtract_entries_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1096
  "distinct (map fst ys) ==> map_of (subtract_entries_raw xs ys) x = (case map_of xs x of None => None | Some v => (case map_of ys x of None => Some v | Some v' => Some (v - v')))"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1097
unfolding subtract_entries_raw_def
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1098
apply (induct ys)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1099
apply auto
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1100
apply (simp split: option.split)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1101
apply (simp add: map_of_map_entry)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1102
apply (auto split: option.split)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1103
apply (metis map_of_eq_None_iff option.simps(3) option.simps(4))
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1104
by (metis map_of_eq_None_iff option.simps(4) option.simps(5))
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1105
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1106
lemma distinct_subtract_entries_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1107
  assumes "distinct (map fst xs)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1108
  shows "distinct (map fst (subtract_entries_raw xs ys))"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1109
using assms
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1110
unfolding subtract_entries_raw_def by (induct ys) (auto simp add: distinct_map_entry)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1111
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1112
text {* Operations on alists *}
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1113
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1114
definition join
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1115
where
46237
99c80c2f841a renamed theory AList to DAList
bulwahn
parents: 46168
diff changeset
  1116
  "join f xs ys = DAList.Alist (join_raw f (DAList.impl_of xs) (DAList.impl_of ys))" 
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1117
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1118
lemma [code abstract]:
46237
99c80c2f841a renamed theory AList to DAList
bulwahn
parents: 46168
diff changeset
  1119
  "DAList.impl_of (join f xs ys) = join_raw f (DAList.impl_of xs) (DAList.impl_of ys)"
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1120
unfolding join_def by (simp add: Alist_inverse distinct_join_raw)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1121
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1122
definition subtract_entries
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1123
where
46237
99c80c2f841a renamed theory AList to DAList
bulwahn
parents: 46168
diff changeset
  1124
  "subtract_entries xs ys = DAList.Alist (subtract_entries_raw (DAList.impl_of xs) (DAList.impl_of ys))"
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1125
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1126
lemma [code abstract]:
46237
99c80c2f841a renamed theory AList to DAList
bulwahn
parents: 46168
diff changeset
  1127
  "DAList.impl_of (subtract_entries xs ys) = subtract_entries_raw (DAList.impl_of xs) (DAList.impl_of ys)"
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1128
unfolding subtract_entries_def by (simp add: Alist_inverse distinct_subtract_entries_raw)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1129
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1130
text {* Implementing multisets by means of association lists *}
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1131
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1132
definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1133
  "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1134
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1135
lemma count_of_multiset:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1136
  "count_of xs \<in> multiset"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1137
proof -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1138
  let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1139
  have "?A \<subseteq> dom (map_of xs)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1140
  proof
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1141
    fix x
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1142
    assume "x \<in> ?A"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1143
    then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1144
    then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1145
    then show "x \<in> dom (map_of xs)" by auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1146
  qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1147
  with finite_dom_map_of [of xs] have "finite ?A"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1148
    by (auto intro: finite_subset)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1149
  then show ?thesis
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
  1150
    by (simp add: count_of_def fun_eq_iff multiset_def)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1151
qed
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1152
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1153
lemma count_simps [simp]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1154
  "count_of [] = (\<lambda>_. 0)"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1155
  "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39301
diff changeset
  1156
  by (simp_all add: count_of_def fun_eq_iff)
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1157
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1158
lemma count_of_empty:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1159
  "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1160
  by (induct xs) (simp_all add: count_of_def)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1161
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1162
lemma count_of_filter:
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1163
  "count_of (List.filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
34943
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1164
  by (induct xs) auto
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1165
46168
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1166
lemma count_of_map_default [simp]:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1167
  "count_of (map_default x b (%x. x + b) xs) y = (if x = y then count_of xs x + b else count_of xs y)"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1168
unfolding count_of_def by (simp add: map_of_map_default split: option.split)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1169
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1170
lemma count_of_join_raw:
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1171
  "distinct (map fst ys) ==> count_of xs x + count_of ys x = count_of (join_raw (%x (x, y). x + y) xs ys) x"
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1172
unfolding count_of_def by (simp add: map_of_join_raw split: option.split)
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1173
bef8c811df20 improving code generation for multisets; adding exhaustive quickcheck generators for multisets
bulwahn
parents: 45989
diff changeset
  1174
lemma count_of_subtract_entries_raw: