src/HOL/Library/Multiset.thy
author blanchet
Thu, 11 Mar 2010 15:33:45 +0100
changeset 35712 77aa29bf14ee
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child 36176 3fe7e97ccca8
permissions -rw-r--r--
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(*  Title:      HOL/Library/Multiset.thy
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    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
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*)
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header {* (Finite) multisets *}
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theory Multiset
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imports Main
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begin
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subsection {* The type of multisets *}
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typedef 'a multiset = "{f :: 'a => nat. finite {x. f x > 0}}"
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  morphisms count Abs_multiset
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proof
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  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
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qed
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lemmas multiset_typedef = Abs_multiset_inverse count_inverse count
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abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
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  "a :# M == 0 < count M a"
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notation (xsymbols)
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  Melem (infix "\<in>#" 50)
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lemma multiset_eq_conv_count_eq:
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  "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
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  by (simp only: count_inject [symmetric] expand_fun_eq)
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lemma multi_count_ext:
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  "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
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  using multiset_eq_conv_count_eq by auto
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text {*
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 \medskip Preservation of the representing set @{term multiset}.
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*}
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lemma const0_in_multiset:
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  "(\<lambda>a. 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma only1_in_multiset:
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  "(\<lambda>b. if b = a then n else 0) \<in> multiset"
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  by (simp add: multiset_def)
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lemma union_preserves_multiset:
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  "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
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  by (simp add: multiset_def)
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lemma diff_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>a. M a - N a) \<in> multiset"
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proof -
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  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemma MCollect_preserves_multiset:
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  assumes "M \<in> multiset"
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  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
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proof -
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  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
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    by auto
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  with assms show ?thesis
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    by (auto simp add: multiset_def intro: finite_subset)
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qed
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lemmas in_multiset = const0_in_multiset only1_in_multiset
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  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
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subsection {* Representing multisets *}
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text {* Multiset comprehension *}
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definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
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  "MCollect M P = Abs_multiset (\<lambda>x. if P x then count M x else 0)"
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syntax
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  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
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translations
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  "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
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text {* Multiset enumeration *}
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instantiation multiset :: (type) "{zero, plus}"
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begin
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definition Mempty_def:
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  "0 = Abs_multiset (\<lambda>a. 0)"
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abbreviation Mempty :: "'a multiset" ("{#}") where
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  "Mempty \<equiv> 0"
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definition union_def:
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  "M + N = Abs_multiset (\<lambda>a. count M a + count N a)"
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instance ..
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end
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definition single :: "'a => 'a multiset" where
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  "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
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syntax
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  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
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translations
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  "{#x, xs#}" == "{#x#} + {#xs#}"
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  "{#x#}" == "CONST single x"
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lemma count_empty [simp]: "count {#} a = 0"
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  by (simp add: Mempty_def in_multiset multiset_typedef)
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
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  by (simp add: single_def in_multiset multiset_typedef)
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subsection {* Basic operations *}
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e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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subsubsection {* Union *}
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lemma count_union [simp]: "count (M + N) a = count M a + count N a"
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  by (simp add: union_def in_multiset multiset_typedef)
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   128
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instance multiset :: (type) cancel_comm_monoid_add proof
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qed (simp_all add: multiset_eq_conv_count_eq)
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e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
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subsubsection {* Difference *}
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instantiation multiset :: (type) minus
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begin
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   137
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definition diff_def:
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  "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"
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   140
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   141
instance ..
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   142
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   143
end
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   144
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   145
lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
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   146
  by (simp add: diff_def in_multiset multiset_typedef)
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   147
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   148
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
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   149
  by (simp add: Mempty_def diff_def in_multiset multiset_typedef)
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
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   152
  by (rule multi_count_ext)
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   153
    (auto simp del: count_single simp add: union_def diff_def in_multiset multiset_typedef)
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   154
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   155
lemma diff_cancel: "A - A = {#}"
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   156
  by (rule multi_count_ext) simp
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   157
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   158
lemma insert_DiffM:
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   159
  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
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   160
  by (clarsimp simp: multiset_eq_conv_count_eq)
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   161
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   162
lemma insert_DiffM2 [simp]:
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  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
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   164
  by (clarsimp simp: multiset_eq_conv_count_eq)
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   165
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   166
lemma diff_right_commute:
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   167
  "(M::'a multiset) - N - Q = M - Q - N"
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   168
  by (auto simp add: multiset_eq_conv_count_eq)
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   169
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   170
lemma diff_union_swap:
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   171
  "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
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   172
  by (auto simp add: multiset_eq_conv_count_eq)
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   173
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   174
lemma diff_union_single_conv:
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   175
  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
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   176
  by (simp add: multiset_eq_conv_count_eq)
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   177
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subsubsection {* Equality of multisets *}
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
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   182
  by (simp add: multiset_eq_conv_count_eq)
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   183
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   184
lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
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   185
  by (auto simp add: multiset_eq_conv_count_eq)
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   186
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   187
lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
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   188
  by (auto simp add: multiset_eq_conv_count_eq)
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   189
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   190
lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
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   191
  by (auto simp add: multiset_eq_conv_count_eq)
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   192
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   193
lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
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   194
  by (auto simp add: multiset_eq_conv_count_eq)
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   195
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   196
lemma diff_single_trivial:
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   197
  "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
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   198
  by (auto simp add: multiset_eq_conv_count_eq)
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   199
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   200
lemma diff_single_eq_union:
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   201
  "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
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   202
  by auto
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   203
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   204
lemma union_single_eq_diff:
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   205
  "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
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   206
  by (auto dest: sym)
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   207
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   208
lemma union_single_eq_member:
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   209
  "M + {#x#} = N \<Longrightarrow> x \<in># N"
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   210
  by auto
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   211
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   212
lemma union_is_single:
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   213
  "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
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   214
proof
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   215
  assume ?rhs then show ?lhs by auto
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   216
next
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   217
  assume ?lhs
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   218
  then have "\<And>b. count (M + N) b = (if b = a then 1 else 0)" by auto
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   219
  then have *: "\<And>b. count M b + count N b = (if b = a then 1 else 0)" by auto
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   220
  then have "count M a + count N a = 1" by auto
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diff changeset
   221
  then have **: "count M a = 1 \<and> count N a = 0 \<or> count M a = 0 \<and> count N a = 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
   222
    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
   223
  from * have "\<And>b. b \<noteq> a \<Longrightarrow> count M b + count N b = 0" 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
   224
  then have ***: "\<And>b. b \<noteq> a \<Longrightarrow> count M b = 0 \<and> count N b = 0" 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
   225
  from ** and *** have
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
    "(\<forall>b. count M b = (if b = a then 1 else 0) \<and> count N b = 0) \<or>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
      (\<forall>b. count M b = 0 \<and> count N 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
   228
    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
   229
  then have
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
    "(\<forall>b. count M b = (if b = a then 1 else 0)) \<and> (\<forall>b. count N b = 0) \<or>
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
      (\<forall>b. count M b = 0) \<and> (\<forall>b. count N 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
   232
    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
   233
  then show ?rhs by (auto simp add: multiset_eq_conv_count_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
   234
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
   235
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   237
  "{#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
   238
  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
   239
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   241
  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?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
   242
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
   243
  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
   244
  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
   245
    (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
   246
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
   247
  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
   248
  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
   249
  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
   250
    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
   251
  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
   252
    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
   253
    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
   254
    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
   255
    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
   256
    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
   257
    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
   258
  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
   259
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
   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
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
   262
  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
   263
   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
   264
  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
   265
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
   266
  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
   267
  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
   268
  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
   269
  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
   270
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
   271
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   273
  "(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
   274
    (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
   275
  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
   276
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   279
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   280
instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   281
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   282
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   283
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
   284
  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
   285
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   286
definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   287
  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
   288
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   289
instance proof
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   290
qed (auto simp add: mset_le_def mset_less_def multiset_eq_conv_count_eq intro: order_trans antisym)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   291
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   292
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
   293
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   294
lemma mset_less_eqI:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   295
  "(\<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
   296
  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
   297
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   298
lemma mset_le_exists_conv:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   299
  "(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
   300
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" 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
   301
apply (auto intro: multiset_eq_conv_count_eq [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
   302
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
   303
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   304
lemma mset_le_mono_add_right_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   305
  "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   306
  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
   307
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   308
lemma mset_le_mono_add_left_cancel [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   309
  "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   310
  by (fact add_le_cancel_left)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   311
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   312
lemma mset_le_mono_add:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   313
  "(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
   314
  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
   315
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   316
lemma mset_le_add_left [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   317
  "(A::'a multiset) \<le> A + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   318
  unfolding mset_le_def by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   319
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   320
lemma mset_le_add_right [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   321
  "B \<le> (A::'a multiset) + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   322
  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
   323
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   324
lemma mset_le_single:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   325
  "a :# B \<Longrightarrow> {#a#} \<le> B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   326
  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
   327
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   328
lemma multiset_diff_union_assoc:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   329
  "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   330
  by (simp add: multiset_eq_conv_count_eq 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
   331
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
lemma mset_le_multiset_union_diff_commute:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   333
  assumes "B \<le> A"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   334
  shows "(A::'a multiset) - B + C = A + C - 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
   335
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
   336
  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
  from this obtain D where "A = B + D" ..
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
  then 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
   339
    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
   340
    apply (subst add_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
   341
    apply (subst multiset_diff_union_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
   342
    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
   343
    apply (simp add: diff_cancel)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   344
    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
   345
    apply (subst 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
   346
    apply (subst multiset_diff_union_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
   347
    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
   348
    apply (simp add: diff_cancel)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
    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
   350
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
   351
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   352
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
   353
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
   354
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
   355
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
   356
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
   357
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   358
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
   359
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
   360
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
   361
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
   362
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
   363
  
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   364
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
   365
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
   366
 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
   367
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
   368
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
   369
 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
   370
 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
   371
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
   372
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   373
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
   374
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
   375
 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
   376
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
   377
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
   378
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   379
lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"
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
   380
  by (auto simp add: mset_less_def mset_le_def multiset_eq_conv_count_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
   381
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   382
lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   383
  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
   384
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   385
lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   386
  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
   387
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   388
lemma mset_less_add_bothsides:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   389
  "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   390
  by (fact add_less_imp_less_right)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   391
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   392
lemma mset_less_empty_nonempty:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   393
  "{#} < S \<longleftrightarrow> S \<noteq> {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   394
  by (auto simp: mset_le_def mset_less_def)
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 mset_less_diff_self:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   397
  "c \<in># B \<Longrightarrow> B - {#c#} < B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   398
  by (auto simp: mset_le_def mset_less_def multiset_eq_conv_count_eq)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   399
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   400
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   401
subsubsection {* Intersection *}
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   402
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   403
instantiation multiset :: (type) semilattice_inf
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   404
begin
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   405
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   406
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
   407
  multiset_inter_def: "inf_multiset A B = A - (A - B)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   408
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   409
instance proof -
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   410
  have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   411
  show "OFCLASS('a multiset, semilattice_inf_class)" proof
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   412
  qed (auto simp add: multiset_inter_def mset_le_def aux)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   413
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   414
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   415
end
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   416
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   417
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
   418
  "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
   419
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   420
lemma multiset_inter_count:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   421
  "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
   422
  by (simp add: multiset_inter_def multiset_typedef)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   423
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   424
lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   425
  by (rule multi_count_ext) (auto simp add: multiset_inter_count)
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
   426
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   427
lemma multiset_union_diff_commute:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   428
  assumes "B #\<inter> C = {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   429
  shows "A + B - C = A - C + B"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   430
proof (rule multi_count_ext)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   431
  fix x
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   432
  from assms have "min (count B x) (count C x) = 0"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   433
    by (auto simp add: multiset_inter_count multiset_eq_conv_count_eq)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   434
  then have "count B x = 0 \<or> count C x = 0"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   435
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   436
  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
   437
    by auto
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   438
qed
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   439
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   440
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   441
subsubsection {* Comprehension (filter) *}
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   442
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   443
lemma count_MCollect [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   444
  "count {# x:#M. P x #} a = (if P a then count M a else 0)"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   445
  by (simp add: MCollect_def in_multiset multiset_typedef)
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   446
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   447
lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   448
  by (rule multi_count_ext) simp
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   449
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   450
lemma MCollect_single [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   451
  "MCollect {#x#} P = (if P x then {#x#} else {#})"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   452
  by (rule multi_count_ext) simp
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   453
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   454
lemma MCollect_union [simp]:
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   455
  "MCollect (M + N) f = MCollect M f + MCollect N f"
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   456
  by (rule multi_count_ext) simp
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   457
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   458
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   459
subsubsection {* Set of elements *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   460
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
   461
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
   462
  "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
   463
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   464
lemma set_of_empty [simp]: "set_of {#} = {}"
26178
nipkow
parents: 26176
diff changeset
   465
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   466
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   467
lemma set_of_single [simp]: "set_of {#b#} = {b}"
26178
nipkow
parents: 26176
diff changeset
   468
by (simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   469
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   470
lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
26178
nipkow
parents: 26176
diff changeset
   471
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   472
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   473
lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
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
   474
by (auto simp add: set_of_def multiset_eq_conv_count_eq)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   475
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   476
lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
26178
nipkow
parents: 26176
diff changeset
   477
by (auto simp add: set_of_def)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   478
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   479
lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
26178
nipkow
parents: 26176
diff changeset
   480
by (auto simp add: set_of_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   481
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
   482
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
   483
  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
   484
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   485
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   486
subsubsection {* Size *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   487
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
   488
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
   489
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
   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
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
   492
  "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
   493
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   495
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   496
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
   497
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   498
lemma size_empty [simp]: "size {#} = 0"
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
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   501
lemma size_single [simp]: "size {#b#} = 1"
26178
nipkow
parents: 26176
diff changeset
   502
by (simp add: size_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   503
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   504
lemma setsum_count_Int:
26178
nipkow
parents: 26176
diff changeset
   505
  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow
parents: 26176
diff changeset
   506
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   507
 apply simp
nipkow
parents: 26176
diff changeset
   508
apply (simp add: Int_insert_left set_of_def)
nipkow
parents: 26176
diff changeset
   509
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   510
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
   511
lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
26178
nipkow
parents: 26176
diff changeset
   512
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   513
apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow
parents: 26176
diff changeset
   514
 prefer 2
nipkow
parents: 26176
diff changeset
   515
 apply (rule ext, simp)
nipkow
parents: 26176
diff changeset
   516
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow
parents: 26176
diff changeset
   517
apply (subst Int_commute)
nipkow
parents: 26176
diff changeset
   518
apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow
parents: 26176
diff changeset
   519
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   520
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   521
lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
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
   522
by (auto simp add: size_def multiset_eq_conv_count_eq)
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   523
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   524
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
26178
nipkow
parents: 26176
diff changeset
   525
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
   526
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   527
lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
26178
nipkow
parents: 26176
diff changeset
   528
apply (unfold size_def)
nipkow
parents: 26176
diff changeset
   529
apply (drule setsum_SucD)
nipkow
parents: 26176
diff changeset
   530
apply auto
nipkow
parents: 26176
diff changeset
   531
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   532
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
   533
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
   534
  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
   535
  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
   536
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
   537
  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
   538
    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
   539
  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
   540
  then show ?thesis by blast
23611
65b168646309 more interpretations
nipkow
parents: 23373
diff changeset
   541
qed
15869
3aca7f05cd12 intersection
kleing
parents: 15867
diff changeset
   542
26016
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   543
f9d1bf2fc59c added multiset comprehension
nipkow
parents: 25759
diff changeset
   544
subsection {* Induction and case splits *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   545
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   546
lemma setsum_decr:
11701
3d51fbf81c17 sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents: 11655
diff changeset
   547
  "finite F ==> (0::nat) < f a ==>
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   548
    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
   549
apply (induct rule: finite_induct)
nipkow
parents: 26176
diff changeset
   550
 apply auto
nipkow
parents: 26176
diff changeset
   551
apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow
parents: 26176
diff changeset
   552
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   553
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   554
lemma rep_multiset_induct_aux:
26178
nipkow
parents: 26176
diff changeset
   555
assumes 1: "P (\<lambda>a. (0::nat))"
nipkow
parents: 26176
diff changeset
   556
  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow
parents: 26176
diff changeset
   557
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow
parents: 26176
diff changeset
   558
apply (unfold multiset_def)
nipkow
parents: 26176
diff changeset
   559
apply (induct_tac n, simp, clarify)
nipkow
parents: 26176
diff changeset
   560
 apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow
parents: 26176
diff changeset
   561
  apply simp
nipkow
parents: 26176
diff changeset
   562
  apply (rule 1)
nipkow
parents: 26176
diff changeset
   563
 apply (rule ext, force, clarify)
nipkow
parents: 26176
diff changeset
   564
apply (frule setsum_SucD, clarify)
nipkow
parents: 26176
diff changeset
   565
apply (rename_tac a)
nipkow
parents: 26176
diff changeset
   566
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow
parents: 26176
diff changeset
   567
 prefer 2
nipkow
parents: 26176
diff changeset
   568
 apply (rule finite_subset)
nipkow
parents: 26176
diff changeset
   569
  prefer 2
nipkow
parents: 26176
diff changeset
   570
  apply assumption
nipkow
parents: 26176
diff changeset
   571
 apply simp
nipkow
parents: 26176
diff changeset
   572
 apply blast
nipkow
parents: 26176
diff changeset
   573
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow
parents: 26176
diff changeset
   574
 prefer 2
nipkow
parents: 26176
diff changeset
   575
 apply (rule ext)
nipkow
parents: 26176
diff changeset
   576
 apply (simp (no_asm_simp))
nipkow
parents: 26176
diff changeset
   577
 apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow
parents: 26176
diff changeset
   578
apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow
parents: 26176
diff changeset
   579
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow
parents: 26176
diff changeset
   580
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow
parents: 26176
diff changeset
   581
 prefer 2
nipkow
parents: 26176
diff changeset
   582
 apply blast
nipkow
parents: 26176
diff changeset
   583
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow
parents: 26176
diff changeset
   584
 prefer 2
nipkow
parents: 26176
diff changeset
   585
 apply blast
nipkow
parents: 26176
diff changeset
   586
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow
parents: 26176
diff changeset
   587
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   588
10313
51e830bb7abe intro_classes by default;
wenzelm
parents: 10277
diff changeset
   589
theorem rep_multiset_induct:
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
   590
  "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
   591
    (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
26178
nipkow
parents: 26176
diff changeset
   592
using rep_multiset_induct_aux by blast
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   593
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   594
theorem multiset_induct [case_names empty add, induct type: multiset]:
26178
nipkow
parents: 26176
diff changeset
   595
assumes empty: "P {#}"
nipkow
parents: 26176
diff changeset
   596
  and add: "!!M x. P M ==> P (M + {#x#})"
nipkow
parents: 26176
diff changeset
   597
shows "P M"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   598
proof -
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   599
  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
   600
  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
   601
  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
   602
    (\<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
   603
  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
   604
    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
   605
    apply (rule count [THEN rep_multiset_induct])
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
   606
     apply (rule empty [unfolded defns])
15072
4861bf6af0b4 new material courtesy of Norbert Voelker
paulson
parents: 14981
diff changeset
   607
    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
   608
     prefer 2
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   609
     apply (simp add: expand_fun_eq)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   610
    apply (erule ssubst)
17200
3a4d03d1a31b tuned presentation;
wenzelm
parents: 17161
diff changeset
   611
    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
   612
    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
   613
    apply (simp add: aux)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   614
    done
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   615
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   616
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   617
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
26178
nipkow
parents: 26176
diff changeset
   618
by (induct M) auto
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   619
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   620
lemma multiset_cases [cases type, case_names empty add]:
26178
nipkow
parents: 26176
diff changeset
   621
assumes em:  "M = {#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   622
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow
parents: 26176
diff changeset
   623
shows "P"
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   624
proof (cases "M = {#}")
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   625
  assume "M = {#}" then show ?thesis using em by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   626
next
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   627
  assume "M \<noteq> {#}"
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   628
  then obtain M' m where "M = M' + {#m#}" 
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   629
    by (blast dest: multi_nonempty_split)
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   630
  then show ?thesis using add by simp
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   631
qed
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   632
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   633
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
26178
nipkow
parents: 26176
diff changeset
   634
apply (cases M)
nipkow
parents: 26176
diff changeset
   635
 apply simp
nipkow
parents: 26176
diff changeset
   636
apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow
parents: 26176
diff changeset
   637
done
25610
72e1563aee09 a fold operation for multisets + more lemmas
kleing
parents: 25595
diff changeset
   638
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
   639
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
   640
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
   641
26033
278025d5282d modified MCollect syntax
nipkow
parents: 26016
diff changeset
   642
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
26178
nipkow
parents: 26176
diff changeset
   643
apply (subst multiset_eq_conv_count_eq)
nipkow
parents: 26176
diff changeset
   644
apply auto
nipkow
parents: 26176
diff changeset
   645
done
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   646
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   647
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
   648
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
   649
  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
   650
  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
   651
  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
   652
    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
   653
  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
   654
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
   655
  case (add S x T)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   656
  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
   657
  have SxsubT: "S + {#x#} < T" by fact
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   658
  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
   659
  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
   660
    by (blast dest: multi_member_split)
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   661
  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
   662
    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
   663
  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
   664
  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
   665
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
   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
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   669
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   671
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   673
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   674
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
   675
  mset_less_rel :: "('a multiset * 'a multiset) set" where
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   676
  "mset_less_rel = {(A,B). A < B}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
   677
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
   678
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
   679
  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
   680
  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
   681
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
   682
  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
   683
    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
   684
  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
   685
  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
   686
    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
   687
  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
   688
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
   689
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   691
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
   692
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
   693
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
   694
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
   695
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   696
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
   697
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
lemma full_multiset_induct [case_names less]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   699
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
   700
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
   701
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
   702
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
   703
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
   704
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
lemma multi_subset_induct [consumes 2, case_names empty add]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   706
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
   707
  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
   708
  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
   709
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
   710
proof -
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   711
  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
   712
  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
   713
  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
   714
    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
   715
  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
   716
    fix x F
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   717
    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
   718
    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
   719
    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
   720
      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
   721
      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
   722
      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
   723
    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
   724
  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
   725
qed
26145
95670b6e1fa3 tuned document;
wenzelm
parents: 26143
diff changeset
   726
17161
57c69627d71a tuned some proofs;
wenzelm
parents: 15869
diff changeset
   727
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
   728
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
   729
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   731
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   733
  "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
   734
  "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
   735
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   736
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
   737
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
   738
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   739
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
   740
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
   741
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set 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
   743
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
   744
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   746
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
   747
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   749
  "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
   750
  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
   751
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   753
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
   754
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
   755
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
   756
 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
   757
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
   758
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
   759
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
   760
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   761
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
   762
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
   763
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   764
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
   765
  "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
   766
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
   767
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
   768
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
   769
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
   770
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
   771
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
   772
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   774
  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set 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
   775
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_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
   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 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
   778
  "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
   779
    (set x = set y) = (multiset_of x = multiset_of 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
   780
by (auto simp: multiset_eq_conv_count_eq 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
   781
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   783
   "(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
   784
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
   785
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
   786
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
   787
      [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
   788
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
   789
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
   790
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   792
  "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
   793
  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
   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 count_filter:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = 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
   797
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
   798
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
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
   800
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
   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
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
   803
 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
   804
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
   805
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#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
   807
by (induct xs) (auto simp add: multiset_eq_conv_count_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
   808
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   809
lemma multiset_of_eq_length:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   810
assumes "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
   811
shows "length xs = length 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
   812
using assms
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
proof (induct arbitrary: ys rule: length_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
   814
  case (1 xs 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
   815
  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
   816
  proof (cases 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
   817
    case Nil with "1.prems" 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
   818
  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
   819
    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
   820
    note xCons = Cons
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   821
    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
   822
    proof (cases 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
   823
      case Nil
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   824
      with "1.prems" Cons 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
   825
    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
   826
      case (Cons y 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
   827
      have x_in_ys: "x = y \<or> x \<in> set 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
   828
      proof (cases "x = 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
   829
        case True then 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
   830
      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
   831
        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
   832
        from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" 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
   833
        with False show ?thesis by (simp add: mem_set_multiset_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
   834
      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
   835
      from "1.hyps" have IH: "length xs' < length xs \<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
   836
        (\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   837
      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#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
   838
        apply -
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
        apply (simp add: multiset_of_remove1, simp only: 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
   840
        apply fastsimp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   841
        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
   842
      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   843
      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" 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
   844
      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   845
    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
   846
  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
   847
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
   848
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   849
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
   850
  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
   851
  @{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
   852
*}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   853
lemma properties_for_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
   854
  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = 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
   855
proof (induct xs arbitrary: 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
   856
  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
   857
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
   858
  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
   859
  then have "x \<in> set 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
   860
    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   861
  with Cons.prems Cons.hyps [of "remove1 x ys"] 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
   862
    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   863
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
   864
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   865
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
   866
  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
   867
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   868
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
   869
  "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
   870
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
   871
  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
   872
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
   873
  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
   874
  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
   875
  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
   876
    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
   877
  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
   878
    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
   879
    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
   880
      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
   881
      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
   882
      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
   883
      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
   884
      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
   885
      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
   886
      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
   887
      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
   888
  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
   889
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
   890
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   891
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
   892
  "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
   893
    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
   894
  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
   895
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
subsubsection {* Association lists -- including rudimentary code generation *}
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and 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
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   899
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
   900
  "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
   901
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   902
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
   903
  "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
   904
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
   905
  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
   906
  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
   907
  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
   908
    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
   909
    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
   910
    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
   911
    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
   912
    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
   913
  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
   914
  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
   915
    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
   916
  then 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
   917
    by (simp add: count_of_def expand_fun_eq 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
   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
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   920
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
   921
  "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
   922
  "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs 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
   923
  by (simp_all add: count_of_def expand_fun_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
   924
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   925
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
   926
  "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
   927
  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
   928
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   929
lemma count_of_filter:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   930
  "count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x 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
   931
  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
   932
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   933
definition Bag :: "('a \<times> nat) 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
   934
  "Bag xs = Abs_multiset (count_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
   935
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   936
code_datatype Bag
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   937
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   938
lemma count_Bag [simp, code]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   939
  "count (Bag xs) = count_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
   940
  by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   941
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   942
lemma Mempty_Bag [code]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   943
  "{#} = Bag []"
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   944
  by (simp add: multiset_eq_conv_count_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
   945
  
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   946
lemma single_Bag [code]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   947
  "{#x#} = Bag [(x, 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
   948
  by (simp add: multiset_eq_conv_count_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
   949
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   950
lemma MCollect_Bag [code]:
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   951
  "MCollect (Bag xs) P = Bag (filter (P \<circ> fst) 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
   952
  by (simp add: multiset_eq_conv_count_eq count_of_filter)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   953
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   954
lemma mset_less_eq_Bag [code]:
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   955
  "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A 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
   956
    (is "?lhs \<longleftrightarrow> ?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
   957
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
   958
  assume ?lhs then 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
   959
    by (auto simp add: mset_le_def count_Bag)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   960
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
   961
  assume ?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
   962
  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
   963
  proof (rule mset_less_eqI)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   964
    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
   965
    from `?rhs` have "count_of xs x \<le> count A 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
   966
      by (cases "x \<in> fst ` set xs") (auto simp add: 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
   967
    then show "count (Bag xs) x \<le> count A 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
   968
      by (simp add: mset_le_def count_Bag)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   969
  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
   970
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
   971
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   972
instantiation multiset :: (eq) 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
   973
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
   974
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   975
definition
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   976
  "HOL.eq A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<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
   977
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   978
instance proof
35268
04673275441a switched notations for pointwise and multiset order
haftmann
parents: 35028
diff changeset
   979
qed (simp add: eq_multiset_def 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
   980
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   981
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
   982
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   983
definition (in term_syntax)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   984
  bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   985
    \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" 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
   986
  [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} 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
   987
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   988
notation fcomp (infixl "o>" 60)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   989
notation scomp (infixl "o\<rightarrow>" 60)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   990
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   991
instantiation multiset :: (random) random
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   992
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
   993
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   994
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
   995
  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>xs. Pair (bagify 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
   996
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   997
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
   998
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
   999
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
  1000
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1001
no_notation fcomp (infixl "o>" 60)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1002
no_notation scomp (infixl "o\<rightarrow>" 60)
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1003
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1004
hide (open) const bagify
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1005
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1006
e97b22500a5c cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
haftmann
parents: 33102
diff changeset
  1007
subsection {* The multiset order *}
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1008
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1009
subsubsection {* Well-foundedness *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1010
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
  1011
definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
  1012
  [code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1013
      (\<forall>b. b :# K --> (b, a) \<in> r)}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1014
28708
a1a436f09ec6 explicit check for pattern discipline before code translation
haftmann
parents: 28562
diff changeset
  1015
definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
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
  1016
  [code del]: "mult r = (mult1 r)\<^sup>+"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1017
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1018
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
26178
nipkow
parents: 26176
diff changeset
  1019
by (simp add: mult1_def)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1020
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1021
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1022
    (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1023
    (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
19582
a669c98b9c24 get rid of 'concl is';
wenzelm
parents: 19564
diff changeset
  1024
  (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1025
proof (unfold mult1_def)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1026
  let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
  1027
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1028
  let ?case1 = "?case1 {(N, M). ?R N M}"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1029
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1030
  assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1031
  then have "\<exists>a' M0' K.
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
  1032
      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1033
  then show "?case1 \<or> ?case2"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1034
  proof (elim exE conjE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1035
    fix a' M0' K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1036
    assume N: "N = M0' + K" and r: "?r K a'"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1037
    assume "M0 + {#a#} = M0' + {#a'#}"
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1038
    then have "M0 = M0' \<and> a = a' \<or>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
  1039
        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1040
      by (simp only: add_eq_conv_ex)
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1041
    then show ?thesis
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1042
    proof (elim disjE conjE exE)
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1043
      assume "M0 = M0'" "a = a'"
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
  1044
      with N r have "?r K a \<and> N = M0 + K" by simp
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1045
      then have ?case2 .. then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1046
    next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1047
      fix K'
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1048
      assume "M0' = K' + {#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
  1049
      with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1050
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1051
      assume "M0 = K' + {#a'#}"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1052
      with r have "?R (K' + K) M0" by blast
18258
836491e9b7d5 tuned induct proofs;
wenzelm
parents: 17778
diff changeset
  1053
      with n have ?case1 by simp then show ?thesis ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1054
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1055
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1056
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1057
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1058
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1059
proof
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1060
  let ?R = "mult1 r"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1061
  let ?W = "acc ?R"
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1062
  {
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1063
    fix M M0 a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1064
    assume M0: "M0 \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1065
      and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1066
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1067
    have "M0 + {#a#} \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1068
    proof (rule accI [of "M0 + {#a#}"])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1069
      fix N
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1070
      assume "(N, M0 + {#a#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1071
      then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1072
          (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1073
        by (rule less_add)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1074
      then show "N \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1075
      proof (elim exE disjE conjE)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1076
        fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1077
        from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1078
        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1079
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1080
      next
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1081
        fix K
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1082
        assume N: "N = M0 + K"
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1083
        assume "\<forall>b. b :# K --> (b, a) \<in> r"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1084
        then have "M0 + K \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1085
        proof (induct K)
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
  1086
          case empty
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1087
          from M0 show "M0 + {#} \<in> ?W" by simp
18730
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
  1088
        next
843da46f89ac tuned proofs;
wenzelm
parents: 18258
diff changeset
  1089
          case (add K x)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1090
          from add.prems have "(x, a) \<in> r" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1091
          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1092
          moreover from add have "M0 + K \<in> ?W" by simp
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1093
          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
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
  1094
          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1095
        qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1096
        then show "N \<in> ?W" by (simp only: N)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1097
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1098
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1099
  } note tedious_reasoning = this
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1100
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1101
  assume wf: "wf r"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1102
  fix M
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1103
  show "M \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1104
  proof (induct M)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1105
    show "{#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1106
    proof (rule accI)
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1107
      fix b assume "(b, {#}) \<in> ?R"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1108
      with not_less_empty show "b \<in> ?W" by contradiction
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1109
    qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1110
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1111
    fix M a assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1112
    from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1113
    proof induct
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1114
      fix a
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1115
      assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1116
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1117
      proof
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1118
        fix M assume "M \<in> ?W"
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1119
        then show "M + {#a#} \<in> ?W"
23373
ead82c82da9e tuned proofs: avoid implicit prems;
wenzelm
parents: 23281
diff changeset
  1120
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1121
      qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1122
    qed
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1123
    from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1124
  qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1125
qed
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1126
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1127
theorem wf_mult1: "wf r ==> wf (mult1 r)"
26178
nipkow
parents: 26176
diff changeset
  1128
by (rule acc_wfI) (rule all_accessible)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1129
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1130
theorem wf_mult: "wf r ==> wf (mult r)"
26178
nipkow
parents: 26176
diff changeset
  1131
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
10249
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1132
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1133
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1134
subsubsection {* Closure-free presentation *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1135
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1136
text {* One direction. *}
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1137
e4d13d8a9011 Multisets (from HOL/Induct/Multiset and friends);
wenzelm
parents:
diff changeset
  1138
lemma mult_implies_one_step:
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1139
  "trans r ==> (M, N) \<in> mult r ==>
11464
ddea204de5bc turned translation for 1::nat into def.
nipkow
parents: 10714
diff changeset
  1140
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
23751
a7c7edf2c5ad Restored set notation.
berghofe
parents: 23611
diff changeset
  1141
    (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
26178
nipkow
parents: 26176
diff changeset
  1142
apply (unfold mult_def mult1_def set_of_def)
nipkow
parents: 26176
diff changeset
  1143
apply (erule converse_trancl_induct, clarify)
nipkow
parents: 26176
diff changeset
  1144
 apply (rule_tac x = M0 in exI, simp, clarify)
nipkow
parents: 26176
diff changeset
  1145
apply (case_tac "a :# K")
nipkow
parents: 26176
diff changeset
  1146
 apply (rule_tac x = I in exI)
nipkow
parents: 26176
diff changeset
  1147
 apply (simp (no_asm))
nipkow
parents: 26176
diff changeset
  1148
 apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
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
  1149
 apply (simp (no_asm_simp) add: add_assoc [symmetric])
26178
nipkow
parents: 26176
diff changeset
  1150
 apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow
parents: 26176
diff changeset
  1151
 apply (simp add: diff_union_single_conv)
nipkow
parents: 26176
diff changeset
  1152
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
  1153
 apply blast
nipkow
parents: 26176
diff changeset
  1154
apply (subgoal_tac "a :# I")
nipkow
parents: 26176
diff changeset
  1155
 apply (rule_tac x = "I - {#a#}" in exI)
nipkow
parents: 26176
diff changeset
  1156
 apply (rule_tac x = "J + {#a#}" in exI)
nipkow
parents: 26176
diff changeset
  1157
 apply (rule_tac x = "K + Ka" in exI)
nipkow
parents: 26176
diff changeset
  1158
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1159
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
  1160
 apply (rule conjI)
nipkow
parents: 26176
diff changeset
  1161
  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow
parents: 26176
diff changeset
  1162
  apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow
parents: 26176
diff changeset
  1163
 apply (simp (no_asm_use) add: trans_def)
nipkow
parents: 26176
diff changeset
  1164
 apply blast
nipkow
parents: 26176
diff changeset
  1165
apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow
parents: 26176