author haftmann Fri, 22 Jan 2010 13:38:28 +0100 changeset 34943 e97b22500a5c parent 34942 d62eddd9e253 child 34944 970e1466028d
cleanup of Multiset.thy: less duplication, tuned and simplified a couple of proofs, less historical organization of sections, conversion from associations lists to multisets, rudimentary code generation
```--- a/src/HOL/Library/Multiset.thy	Thu Jan 21 09:27:57 2010 +0100
+++ b/src/HOL/Library/Multiset.thy	Fri Jan 22 13:38:28 2010 +0100
@@ -2,34 +2,21 @@
Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
*)

-header {* Multisets *}
+header {* (Finite) multisets *}

theory Multiset
-imports List Main
+imports Main
begin

subsection {* The type of multisets *}

-typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
+typedef 'a multiset = "{f :: 'a => nat. finite {x. f x > 0}}"
+  morphisms count Abs_multiset
proof
show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
qed

-lemmas multiset_typedef [simp] =
-    Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
-  and [simp] = Rep_multiset_inject [symmetric]
-
-definition Mempty :: "'a multiset"  ("{#}") where
-  [code del]: "{#} = Abs_multiset (\<lambda>a. 0)"
-
-definition single :: "'a => 'a multiset" where
-  [code del]: "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
-
-definition count :: "'a multiset => 'a => nat" where
-  "count = Rep_multiset"
-
-definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
-  "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
+lemmas multiset_typedef = Abs_multiset_inverse count_inverse count

abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
"a :# M == 0 < count M a"
@@ -37,142 +24,456 @@
notation (xsymbols)
Melem (infix "\<in>#" 50)

+lemma multiset_eq_conv_count_eq:
+  "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
+  by (simp only: count_inject [symmetric] expand_fun_eq)
+
+lemma multi_count_ext:
+  "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
+  using multiset_eq_conv_count_eq by auto
+
+text {*
+ \medskip Preservation of the representing set @{term multiset}.
+*}
+
+lemma const0_in_multiset:
+  "(\<lambda>a. 0) \<in> multiset"
+  by (simp add: multiset_def)
+
+lemma only1_in_multiset:
+  "(\<lambda>b. if b = a then n else 0) \<in> multiset"
+  by (simp add: multiset_def)
+
+lemma union_preserves_multiset:
+  "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"
+  by (simp add: multiset_def)
+
+lemma diff_preserves_multiset:
+  assumes "M \<in> multiset"
+  shows "(\<lambda>a. M a - N a) \<in> multiset"
+proof -
+  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"
+    by auto
+  with assms show ?thesis
+    by (auto simp add: multiset_def intro: finite_subset)
+qed
+
+lemma MCollect_preserves_multiset:
+  assumes "M \<in> multiset"
+  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"
+proof -
+  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"
+    by auto
+  with assms show ?thesis
+    by (auto simp add: multiset_def intro: finite_subset)
+qed
+
+lemmas in_multiset = const0_in_multiset only1_in_multiset
+  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
+
+
+subsection {* Representing multisets *}
+
+text {* Multiset comprehension *}
+
+definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
+  "MCollect M P = Abs_multiset (\<lambda>x. if P x then count M x else 0)"
+
syntax
"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
translations
"{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"

-definition set_of :: "'a multiset => 'a set" where
-  "set_of M = {x. x :# M}"

-instantiation multiset :: (type) "{plus, minus, zero, size}"
+text {* Multiset enumeration *}
+
+instantiation multiset :: (type) "{zero, plus}"
begin

-definition union_def [code del]:
-  "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
-
-definition diff_def [code del]:
-  "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
+definition Mempty_def:
+  "0 = Abs_multiset (\<lambda>a. 0)"

-definition Zero_multiset_def [simp]:
-  "0 = {#}"
+abbreviation Mempty :: "'a multiset" ("{#}") where
+  "Mempty \<equiv> 0"

-definition size_def:
-  "size M = setsum (count M) (set_of M)"
+definition union_def:
+  "M + N = Abs_multiset (\<lambda>a. count M a + count N a)"

instance ..

end

-definition
-  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
-  "multiset_inter A B = A - (A - B)"
+definition single :: "'a => 'a multiset" where
+  "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"

-text {* Multiset Enumeration *}
syntax
"_multiset" :: "args => 'a multiset"    ("{#(_)#}")
translations
"{#x, xs#}" == "{#x#} + {#xs#}"
"{#x#}" == "CONST single x"

-
-text {*
- \medskip Preservation of the representing set @{term multiset}.
-*}
+lemma count_empty [simp]: "count {#} a = 0"
+  by (simp add: Mempty_def in_multiset multiset_typedef)

-lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
-by (simp add: multiset_def)
-
-lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
-by (simp add: multiset_def)
-
-lemma union_preserves_multiset:
-  "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
-by (simp add: multiset_def)
+lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
+  by (simp add: single_def in_multiset multiset_typedef)

-lemma diff_preserves_multiset:
-  "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
-apply (simp add: multiset_def)
-apply (rule finite_subset)
- apply auto
-done
-
-lemma MCollect_preserves_multiset:
-  "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
-apply (simp add: multiset_def)
-apply (rule finite_subset, auto)
-done
-
-lemmas in_multiset = const0_in_multiset only1_in_multiset
-  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
-
-
-subsection {* Algebraic properties *}
+subsection {* Basic operations *}

subsubsection {* Union *}

-lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
-by (simp add: union_def Mempty_def in_multiset)
-
-lemma union_commute: "M + N = N + (M::'a multiset)"
-by (simp add: union_def add_ac in_multiset)
-
-lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
-by (simp add: union_def add_ac in_multiset)
+lemma count_union [simp]: "count (M + N) a = count M a + count N a"
+  by (simp add: union_def in_multiset multiset_typedef)

-lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
-proof -
-  have "M + (N + K) = (N + K) + M" by (rule union_commute)
-  also have "\<dots> = N + (K + M)" by (rule union_assoc)
-  also have "K + M = M + K" by (rule union_commute)
-  finally show ?thesis .
-qed
-
-lemmas union_ac = union_assoc union_commute union_lcomm
-
-instance multiset :: (type) comm_monoid_add
-proof
-  fix a b c :: "'a multiset"
-  show "(a + b) + c = a + (b + c)" by (rule union_assoc)
-  show "a + b = b + a" by (rule union_commute)
-  show "0 + a = a" by simp
-qed
+instance multiset :: (type) cancel_comm_monoid_add proof
+qed (simp_all add: multiset_eq_conv_count_eq)

subsubsection {* Difference *}

+instantiation multiset :: (type) minus
+begin
+
+definition diff_def:
+  "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"
+
+instance ..
+
+end
+
+lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
+  by (simp add: diff_def in_multiset multiset_typedef)
+
lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
-by (simp add: Mempty_def diff_def in_multiset)
+  by (simp add: Mempty_def diff_def in_multiset multiset_typedef)

lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
-by (simp add: union_def diff_def in_multiset)
+  by (rule multi_count_ext)
+    (auto simp del: count_single simp add: union_def diff_def in_multiset multiset_typedef)

lemma diff_cancel: "A - A = {#}"
-by (simp add: diff_def Mempty_def)
+  by (rule multi_count_ext) simp
+
+lemma insert_DiffM:
+  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
+  by (clarsimp simp: multiset_eq_conv_count_eq)
+
+lemma insert_DiffM2 [simp]:
+  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
+  by (clarsimp simp: multiset_eq_conv_count_eq)
+
+lemma diff_right_commute:
+  "(M::'a multiset) - N - Q = M - Q - N"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma diff_union_swap:
+  "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma diff_union_single_conv:
+  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
+  by (simp add: multiset_eq_conv_count_eq)

-subsubsection {* Count of elements *}
+subsubsection {* Intersection *}
+
+definition multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
+  "multiset_inter A B = A - (A - B)"
+
+lemma multiset_inter_count:
+  "count (A #\<inter> B) x = min (count A x) (count B x)"
+  by (simp add: multiset_inter_def multiset_typedef)

-lemma count_empty [simp]: "count {#} a = 0"
-by (simp add: count_def Mempty_def in_multiset)
+lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
+  by (rule multi_count_ext) (simp add: multiset_inter_count)
+
+lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
+  by (rule multi_count_ext) (simp add: multiset_inter_count)
+
+lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
+  by (rule multi_count_ext) (simp add: multiset_inter_count)

-lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
-by (simp add: count_def single_def in_multiset)
+lemmas multiset_inter_ac =
+  multiset_inter_commute
+  multiset_inter_assoc
+  multiset_inter_left_commute
+
+lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
+  by (rule multi_count_ext) (auto simp add: multiset_inter_count)

-lemma count_union [simp]: "count (M + N) a = count M a + count N a"
-by (simp add: count_def union_def in_multiset)
+lemma multiset_union_diff_commute:
+  assumes "B #\<inter> C = {#}"
+  shows "A + B - C = A - C + B"
+proof (rule multi_count_ext)
+  fix x
+  from assms have "min (count B x) (count C x) = 0"
+    by (auto simp add: multiset_inter_count multiset_eq_conv_count_eq)
+  then have "count B x = 0 \<or> count C x = 0"
+    by auto
+  then show "count (A + B - C) x = count (A - C + B) x"
+    by auto
+qed

-lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
-by (simp add: count_def diff_def in_multiset)
+
+subsubsection {* Comprehension (filter) *}

lemma count_MCollect [simp]:
"count {# x:#M. P x #} a = (if P a then count M a else 0)"
-by (simp add: count_def MCollect_def in_multiset)
+  by (simp add: MCollect_def in_multiset multiset_typedef)
+
+lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
+  by (rule multi_count_ext) simp
+
+lemma MCollect_single [simp]:
+  "MCollect {#x#} P = (if P x then {#x#} else {#})"
+  by (rule multi_count_ext) simp
+
+lemma MCollect_union [simp]:
+  "MCollect (M + N) f = MCollect M f + MCollect N f"
+  by (rule multi_count_ext) simp
+
+
+subsubsection {* Equality of multisets *}
+
+lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
+  by (simp add: multiset_eq_conv_count_eq)
+
+lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma diff_single_trivial:
+  "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
+  by (auto simp add: multiset_eq_conv_count_eq)
+
+lemma diff_single_eq_union:
+  "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"
+  by auto
+
+lemma union_single_eq_diff:
+  "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"
+  by (auto dest: sym)
+
+lemma union_single_eq_member:
+  "M + {#x#} = N \<Longrightarrow> x \<in># N"
+  by auto
+
+lemma union_is_single:
+  "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")
+proof
+  assume ?rhs then show ?lhs by auto
+next
+  assume ?lhs
+  then have "\<And>b. count (M + N) b = (if b = a then 1 else 0)" by auto
+  then have *: "\<And>b. count M b + count N b = (if b = a then 1 else 0)" by auto
+  then have "count M a + count N a = 1" by auto
+  then have **: "count M a = 1 \<and> count N a = 0 \<or> count M a = 0 \<and> count N a = 1"
+    by auto
+  from * have "\<And>b. b \<noteq> a \<Longrightarrow> count M b + count N b = 0" by auto
+  then have ***: "\<And>b. b \<noteq> a \<Longrightarrow> count M b = 0 \<and> count N b = 0" by auto
+  from ** and *** have
+    "(\<forall>b. count M b = (if b = a then 1 else 0) \<and> count N b = 0) \<or>
+      (\<forall>b. count M b = 0 \<and> count N b = (if b = a then 1 else 0))"
+    by auto
+  then have
+    "(\<forall>b. count M b = (if b = a then 1 else 0)) \<and> (\<forall>b. count N b = 0) \<or>
+      (\<forall>b. count M b = 0) \<and> (\<forall>b. count N b = (if b = a then 1 else 0))"
+    by auto
+  then show ?rhs by (auto simp add: multiset_eq_conv_count_eq)
+qed
+
+lemma single_is_union:
+  "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"
+  by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)
+
+lemma add_eq_conv_diff:
+  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")
+proof
+  assume ?rhs then show ?lhs
+  by (auto simp add: add_assoc add_commute [of "{#b#}"])
+    (drule sym, simp add: add_assoc [symmetric])
+next
+  assume ?lhs
+  show ?rhs
+  proof (cases "a = b")
+    case True with `?lhs` show ?thesis by simp
+  next
+    case False
+    from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)
+    with False have "a \<in># N" by auto
+    moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
+    moreover note False
+    ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
+  qed
+qed
+
+lemma insert_noteq_member:
+  assumes BC: "B + {#b#} = C + {#c#}"
+   and bnotc: "b \<noteq> c"
+  shows "c \<in># B"
+proof -
+  have "c \<in># C + {#c#}" by simp
+  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
+  then have "c \<in># B + {#b#}" using BC by simp
+  then show "c \<in># B" using nc by simp
+qed
+
+lemma add_eq_conv_ex:
+  "(M + {#a#} = N + {#b#}) =
+    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
+  by (auto simp add: add_eq_conv_diff)
+
+
+subsubsection {* Pointwise ordering induced by count *}
+
+definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
+  "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
+
+definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
+  "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
+
+notation mset_le  (infix "\<subseteq>#" 50)
+notation mset_less  (infix "\<subset>#" 50)
+
+lemma mset_less_eqI:
+  "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<subseteq># B"
+  by (simp add: mset_le_def)
+
+lemma mset_le_refl[simp]: "A \<le># A"
+unfolding mset_le_def by auto
+
+lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
+unfolding mset_le_def by (fast intro: order_trans)
+
+lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
+apply (unfold mset_le_def)
+apply (rule multiset_eq_conv_count_eq [THEN iffD2])
+apply (blast intro: order_antisym)
+done
+
+lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
+apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
+apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
+done
+
+lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
+unfolding mset_le_def by auto
+
+lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
+unfolding mset_le_def by auto
+
+lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
+apply (unfold mset_le_def)
+apply auto
+apply (erule_tac x = a in allE)+
+apply auto
+done
+
+lemma mset_le_add_left[simp]: "A \<le># A + B"
+unfolding mset_le_def by auto
+
+lemma mset_le_add_right[simp]: "B \<le># A + B"
+unfolding mset_le_def by auto
+
+lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
+by (simp add: mset_le_def)
+
+lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
+by (simp add: multiset_eq_conv_count_eq mset_le_def)
+
+lemma mset_le_multiset_union_diff_commute:
+assumes "B \<le># A"
+shows "A - B + C = A + C - B"
+proof -
+  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
+  from this obtain D where "A = B + D" ..
+  then show ?thesis
+    apply simp
+    apply (subst add_commute)
+    apply (subst multiset_diff_union_assoc)
+    apply simp
+    apply (simp add: diff_cancel)
+    apply (subst add_assoc)
+    apply (subst add_commute [of "B" _])
+    apply (subst multiset_diff_union_assoc)
+    apply simp
+    apply (simp add: diff_cancel)
+    done
+qed
+
+interpretation mset_order: order "op \<le>#" "op <#"
+proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
+  mset_le_trans simp: mset_less_def)
+
+interpretation mset_order_cancel_semigroup:
+  pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
+proof qed (erule mset_le_mono_add [OF mset_le_refl])
+
+interpretation mset_order_semigroup_cancel:
+  pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
+proof qed simp
+
+lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
+apply (clarsimp simp: mset_le_def mset_less_def)
+apply (erule_tac x=x in allE)
+apply auto
+done
+
+lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
+apply (clarsimp simp: mset_le_def mset_less_def)
+apply (erule_tac x = x in allE)
+apply auto
+done
+
+lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
+apply (rule conjI)
+ apply (simp add: mset_lessD)
+apply (clarsimp simp: mset_le_def mset_less_def)
+apply safe
+ apply (erule_tac x = a in allE)
+ apply (auto split: split_if_asm)
+done
+
+lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
+apply (rule conjI)
+ apply (simp add: mset_leD)
+apply (force simp: mset_le_def mset_less_def split: split_if_asm)
+done
+
+lemma mset_less_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False"
+  by (auto simp add: mset_less_def mset_le_def multiset_eq_conv_count_eq)
+
+lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
+by (auto simp: mset_le_def mset_less_def)
+
+lemma multi_psub_self[simp]: "A \<subset># A = False"
+by (auto simp: mset_le_def mset_less_def)
+
+lemma mset_less_add_bothsides:
+  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
+by (auto simp: mset_le_def mset_less_def)
+
+lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
+by (auto simp: mset_le_def mset_less_def)
+
+lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
+  by (auto simp: mset_le_def mset_less_def multiset_eq_conv_count_eq)

subsubsection {* Set of elements *}

+definition set_of :: "'a multiset => 'a set" where
+  "set_of M = {x. x :# M}"
+
lemma set_of_empty [simp]: "set_of {#} = {}"
by (simp add: set_of_def)

@@ -183,7 +484,7 @@
by (auto simp add: set_of_def)

lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
-by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
+by (auto simp add: set_of_def multiset_eq_conv_count_eq)

lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
by (auto simp add: set_of_def)
@@ -191,18 +492,28 @@
lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
by (auto simp add: set_of_def)

+lemma finite_set_of [iff]: "finite (set_of M)"
+  using count [of M] by (simp add: multiset_def set_of_def)
+

subsubsection {* Size *}

+instantiation multiset :: (type) size
+begin
+
+definition size_def:
+  "size M = setsum (count M) (set_of M)"
+
+instance ..
+
+end
+
lemma size_empty [simp]: "size {#} = 0"
by (simp add: size_def)

lemma size_single [simp]: "size {#b#} = 1"
by (simp add: size_def)

-lemma finite_set_of [iff]: "finite (set_of M)"
-using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
-
lemma setsum_count_Int:
"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
apply (induct rule: finite_induct)
@@ -221,9 +532,7 @@
done

lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
-apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
-apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
-done
+by (auto simp add: size_def multiset_eq_conv_count_eq)

lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
@@ -234,149 +543,16 @@
apply auto
done

-
-subsubsection {* Equality of multisets *}
-
-lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
-by (simp add: count_def expand_fun_eq)
-
-lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
-by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
-
-lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
-by (auto simp add: single_def in_multiset expand_fun_eq)
-
-lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
-by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
-
-lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
-by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
-
-lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
-by (simp add: union_def in_multiset expand_fun_eq)
-
-lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
-by (simp add: union_def in_multiset expand_fun_eq)
-
-lemma union_is_single:
-  "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
-apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
-apply blast
-done
-
-lemma single_is_union:
-  "({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
-apply (unfold Mempty_def single_def union_def)
-apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
-apply (blast dest: sym)
-done
-
-lemma add_eq_conv_diff:
-  "(M + {#a#} = N + {#b#}) =
-   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
-using [[simproc del: neq]]
-apply (unfold single_def union_def diff_def)
-apply (simp (no_asm) add: in_multiset expand_fun_eq)
-apply (rule conjI, force, safe, simp_all)
-apply (simp add: eq_sym_conv)
-done
-
-declare Rep_multiset_inject [symmetric, simp del]
-
-instance multiset :: (type) cancel_ab_semigroup_add
-proof
-  fix a b c :: "'a multiset"
-  show "a + b = a + c \<Longrightarrow> b = c" by simp
+lemma size_eq_Suc_imp_eq_union:
+  assumes "size M = Suc n"
+  shows "\<exists>a N. M = N + {#a#}"
+proof -
+  from assms obtain a where "a \<in># M"
+    by (erule size_eq_Suc_imp_elem [THEN exE])
+  then have "M = M - {#a#} + {#a#}" by simp
+  then show ?thesis by blast
qed

-lemma insert_DiffM:
-  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
-by (clarsimp simp: multiset_eq_conv_count_eq)
-
-lemma insert_DiffM2[simp]:
-  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
-by (clarsimp simp: multiset_eq_conv_count_eq)
-
-lemma multi_union_self_other_eq:
-  "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
-by (induct A arbitrary: X Y) auto
-
-lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
-by (metis single_not_empty union_empty union_left_cancel)
-
-lemma insert_noteq_member:
-  assumes BC: "B + {#b#} = C + {#c#}"
-   and bnotc: "b \<noteq> c"
-  shows "c \<in># B"
-proof -
-  have "c \<in># C + {#c#}" by simp
-  have nc: "\<not> c \<in># {#b#}" using bnotc by simp
-  then have "c \<in># B + {#b#}" using BC by simp
-  then show "c \<in># B" using nc by simp
-qed
-
-
-lemma add_eq_conv_ex:
-  "(M + {#a#} = N + {#b#}) =
-    (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
-by (auto simp add: add_eq_conv_diff)
-
-
-lemma empty_multiset_count:
-  "(\<forall>x. count A x = 0) = (A = {#})"
-by (metis count_empty multiset_eq_conv_count_eq)
-
-
-subsubsection {* Intersection *}
-
-lemma multiset_inter_count:
-  "count (A #\<inter> B) x = min (count A x) (count B x)"
-by (simp add: multiset_inter_def)
-
-lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
-by (simp add: multiset_eq_conv_count_eq multiset_inter_count
-    min_max.inf_commute)
-
-lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
-by (simp add: multiset_eq_conv_count_eq multiset_inter_count
-    min_max.inf_assoc)
-
-lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
-by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
-
-lemmas multiset_inter_ac =
-  multiset_inter_commute
-  multiset_inter_assoc
-  multiset_inter_left_commute
-
-lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
-by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
-
-lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
-apply (simp add: multiset_eq_conv_count_eq multiset_inter_count
-    split: split_if_asm)
-apply clarsimp
-apply (erule_tac x = a in allE)
-apply auto
-done
-
-
-subsubsection {* Comprehension (filter) *}
-
-lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
-by (simp add: MCollect_def Mempty_def Abs_multiset_inject
-    in_multiset expand_fun_eq)
-
-lemma MCollect_single [simp]:
-  "MCollect {#x#} P = (if P x then {#x#} else {#})"
-by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
-    in_multiset expand_fun_eq)
-
-lemma MCollect_union [simp]:
-  "MCollect (M+N) f = MCollect M f + MCollect N f"
-by (simp add: MCollect_def union_def Abs_multiset_inject
-    in_multiset expand_fun_eq)
-

subsection {* Induction and case splits *}

@@ -434,17 +610,20 @@
shows "P M"
proof -
note defns = union_def single_def Mempty_def
+  note add' = add [unfolded defns, simplified]
+  have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) =
+    (\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset)
show ?thesis
-    apply (rule Rep_multiset_inverse [THEN subst])
-    apply (rule Rep_multiset [THEN rep_multiset_induct])
+    apply (rule count_inverse [THEN subst])
+    apply (rule count [THEN rep_multiset_induct])
apply (rule empty [unfolded defns])
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
prefer 2
apply (simp add: expand_fun_eq)
apply (erule ssubst)
apply (erule Abs_multiset_inverse [THEN subst])
-    apply (drule add [unfolded defns, simplified])
-    apply(simp add:in_multiset)
+    apply (drule add')
+    apply (simp add: aux)
done
qed

@@ -470,18 +649,379 @@
apply (rule_tac x="M - {#x#}" in exI, simp)
done

+lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
+by (cases "B = {#}") (auto dest: multi_member_split)
+
lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
apply (subst multiset_eq_conv_count_eq)
apply auto
done

-declare multiset_typedef [simp del]
+lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
+proof (induct A arbitrary: B)
+  case (empty M)
+  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
+  then obtain M' x where "M = M' + {#x#}"
+    by (blast dest: multi_nonempty_split)
+  then show ?case by simp
+next
+  case (add S x T)
+  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
+  have SxsubT: "S + {#x#} \<subset># T" by fact
+  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
+  then obtain T' where T: "T = T' + {#x#}"
+    by (blast dest: multi_member_split)
+  then have "S \<subset># T'" using SxsubT
+    by (blast intro: mset_less_add_bothsides)
+  then have "size S < size T'" using IH by simp
+  then show ?case using T by simp
+qed
+
+
+subsubsection {* Strong induction and subset induction for multisets *}
+
+text {* Well-foundedness of proper subset operator: *}
+
+text {* proper multiset subset *}
+
+definition
+  mset_less_rel :: "('a multiset * 'a multiset) set" where
+  "mset_less_rel = {(A,B). A \<subset># B}"

-lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
-by (cases "B = {#}") (auto dest: multi_member_split)
+lemma multiset_add_sub_el_shuffle:
+  assumes "c \<in># B" and "b \<noteq> c"
+  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
+proof -
+  from `c \<in># B` obtain A where B: "B = A + {#c#}"
+    by (blast dest: multi_member_split)
+  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
+  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
+    by (simp add: add_ac)
+  then show ?thesis using B by simp
+qed
+
+lemma wf_mset_less_rel: "wf mset_less_rel"
+apply (unfold mset_less_rel_def)
+apply (rule wf_measure [THEN wf_subset, where f1=size])
+apply (clarsimp simp: measure_def inv_image_def mset_less_size)
+done
+
+text {* The induction rules: *}
+
+lemma full_multiset_induct [case_names less]:
+assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
+shows "P B"
+apply (rule wf_mset_less_rel [THEN wf_induct])
+apply (rule ih, auto simp: mset_less_rel_def)
+done
+
+lemma multi_subset_induct [consumes 2, case_names empty add]:
+assumes "F \<subseteq># A"
+  and empty: "P {#}"
+  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
+shows "P F"
+proof -
+  from `F \<subseteq># A`
+  show ?thesis
+  proof (induct F)
+    show "P {#}" by fact
+  next
+    fix x F
+    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
+    show "P (F + {#x#})"
+    proof (rule insert)
+      from i show "x \<in># A" by (auto dest: mset_le_insertD)
+      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
+      with P show "P F" .
+    qed
+  qed
+qed

-subsection {* Orderings *}
+subsection {* Alternative representations *}
+
+subsubsection {* Lists *}
+
+primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
+  "multiset_of [] = {#}" |
+  "multiset_of (a # x) = multiset_of x + {# a #}"
+
+lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
+by (induct x) auto
+
+lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
+by (induct x) auto
+
+lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
+by (induct x) auto
+
+lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
+by (induct xs) auto
+
+lemma multiset_of_append [simp]:
+  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
+  by (induct xs arbitrary: ys) (auto simp: add_ac)
+
+lemma surj_multiset_of: "surj multiset_of"
+apply (unfold surj_def)
+apply (rule allI)
+apply (rule_tac M = y in multiset_induct)
+ apply auto
+apply (rule_tac x = "x # xa" in exI)
+apply auto
+done
+
+lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
+by (induct x) auto
+
+lemma distinct_count_atmost_1:
+  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
+apply (induct x, simp, rule iffI, simp_all)
+apply (rule conjI)
+apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
+apply (erule_tac x = a in allE, simp, clarify)
+apply (erule_tac x = aa in allE, simp)
+done
+
+lemma multiset_of_eq_setD:
+  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
+by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
+
+lemma set_eq_iff_multiset_of_eq_distinct:
+  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
+    (set x = set y) = (multiset_of x = multiset_of y)"
+by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
+
+lemma set_eq_iff_multiset_of_remdups_eq:
+   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
+apply (rule iffI)
+apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
+apply (drule distinct_remdups [THEN distinct_remdups
+      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
+apply simp
+done
+
+lemma multiset_of_compl_union [simp]:
+  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
+  by (induct xs) (auto simp: add_ac)
+
+lemma count_filter:
+  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
+by (induct xs) auto
+
+lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
+apply (induct ls arbitrary: i)
+ apply simp
+apply (case_tac i)
+ apply auto
+done
+
+lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
+by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
+
+lemma multiset_of_eq_length:
+assumes "multiset_of xs = multiset_of ys"
+shows "length xs = length ys"
+using assms
+proof (induct arbitrary: ys rule: length_induct)
+  case (1 xs ys)
+  show ?case
+  proof (cases xs)
+    case Nil with "1.prems" show ?thesis by simp
+  next
+    case (Cons x xs')
+    note xCons = Cons
+    show ?thesis
+    proof (cases ys)
+      case Nil
+      with "1.prems" Cons show ?thesis by simp
+    next
+      case (Cons y ys')
+      have x_in_ys: "x = y \<or> x \<in> set ys'"
+      proof (cases "x = y")
+        case True then show ?thesis ..
+      next
+        case False
+        from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
+        with False show ?thesis by (simp add: mem_set_multiset_eq)
+      qed
+      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
+        (\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
+      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
+        apply -
+        apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
+        apply fastsimp
+        done
+      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
+      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
+      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
+    qed
+  qed
+qed
+
+text {*
+  This lemma shows which properties suffice to show that a function
+  @{text "f"} with @{text "f xs = ys"} behaves like sort.
+*}
+lemma properties_for_sort:
+  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
+proof (induct xs arbitrary: ys)
+  case Nil then show ?case by simp
+next
+  case (Cons x xs)
+  then have "x \<in> set ys"
+    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
+  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
+    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
+qed
+
+lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
+apply (induct xs)
+ apply auto
+apply (rule mset_le_trans)
+ apply auto
+done
+
+lemma multiset_of_update:
+  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
+proof (induct ls arbitrary: i)
+  case Nil then show ?case by simp
+next
+  case (Cons x xs)
+  show ?case
+  proof (cases i)
+    case 0 then show ?thesis by simp
+  next
+    case (Suc i')
+    with Cons show ?thesis
+      apply simp
+      apply (subst add_assoc)
+      apply (subst add_commute [of "{#v#}" "{#x#}"])
+      apply (subst add_assoc [symmetric])
+      apply simp
+      apply (rule mset_le_multiset_union_diff_commute)
+      apply (simp add: mset_le_single nth_mem_multiset_of)
+      done
+  qed
+qed
+
+lemma multiset_of_swap:
+  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
+    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
+  by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)
+
+
+subsubsection {* Association lists -- including rudimentary code generation *}
+
+definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where
+  "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"
+
+lemma count_of_multiset:
+  "count_of xs \<in> multiset"
+proof -
+  let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"
+  have "?A \<subseteq> dom (map_of xs)"
+  proof
+    fix x
+    assume "x \<in> ?A"
+    then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp
+    then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto
+    then show "x \<in> dom (map_of xs)" by auto
+  qed
+  with finite_dom_map_of [of xs] have "finite ?A"
+    by (auto intro: finite_subset)
+  then show ?thesis
+    by (simp add: count_of_def expand_fun_eq multiset_def)
+qed
+
+lemma count_simps [simp]:
+  "count_of [] = (\<lambda>_. 0)"
+  "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
+  by (simp_all add: count_of_def expand_fun_eq)
+
+lemma count_of_empty:
+  "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"
+  by (induct xs) (simp_all add: count_of_def)
+
+lemma count_of_filter:
+  "count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
+  by (induct xs) auto
+
+definition Bag :: "('a \<times> nat) list \<Rightarrow> 'a multiset" where
+  "Bag xs = Abs_multiset (count_of xs)"
+
+code_datatype Bag
+
+lemma count_Bag [simp, code]:
+  "count (Bag xs) = count_of xs"
+  by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)
+
+lemma Mempty_Bag [code]:
+  "{#} = Bag []"
+  by (simp add: multiset_eq_conv_count_eq)
+
+lemma single_Bag [code]:
+  "{#x#} = Bag [(x, 1)]"
+  by (simp add: multiset_eq_conv_count_eq)
+
+lemma MCollect_Bag [code]:
+  "MCollect (Bag xs) P = Bag (filter (P \<circ> fst) xs)"
+  by (simp add: multiset_eq_conv_count_eq count_of_filter)
+
+lemma mset_less_eq_Bag [code]:
+  "Bag xs \<subseteq># A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A x)"
+    (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+  assume ?lhs then show ?rhs
+    by (auto simp add: mset_le_def count_Bag)
+next
+  assume ?rhs
+  show ?lhs
+  proof (rule mset_less_eqI)
+    fix x
+    from `?rhs` have "count_of xs x \<le> count A x"
+      by (cases "x \<in> fst ` set xs") (auto simp add: count_of_empty)
+    then show "count (Bag xs) x \<le> count A x"
+      by (simp add: mset_le_def count_Bag)
+  qed
+qed
+
+instantiation multiset :: (eq) eq
+begin
+
+definition
+  "HOL.eq A B \<longleftrightarrow> A \<subseteq># B \<and> B \<subseteq># A"
+
+instance proof
+qed (simp add: eq_multiset_def mset_order.eq_iff)
+
+end
+
+definition (in term_syntax)
+  bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term)
+    \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
+  [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"
+
+notation fcomp (infixl "o>" 60)
+notation scomp (infixl "o\<rightarrow>" 60)
+
+instantiation multiset :: (random) random
+begin
+
+definition
+  "Quickcheck.random i = Quickcheck.random i o\<rightarrow> (\<lambda>xs. Pair (bagify xs))"
+
+instance ..
+
+end
+
+no_notation fcomp (infixl "o>" 60)
+no_notation scomp (infixl "o\<rightarrow>" 60)
+
+hide (open) const bagify
+
+
+subsection {* The multiset order *}

subsubsection {* Well-foundedness *}

@@ -490,7 +1030,7 @@
(\<forall>b. b :# K --> (b, a) \<in> r)}"

definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
-  "mult r = (mult1 r)\<^sup>+"
+  [code del]: "mult r = (mult1 r)\<^sup>+"

lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
by (simp add: mult1_def)
@@ -523,7 +1063,7 @@
next
fix K'
assume "M0' = K' + {#a#}"
-      with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
+      with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)

assume "M0 = K' + {#a'#}"
with r have "?R (K' + K) M0" by blast
@@ -568,7 +1108,7 @@
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
moreover from add have "M0 + K \<in> ?W" by simp
ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
-          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
+          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)
qed
then show "N \<in> ?W" by (simp only: N)
qed
@@ -610,11 +1150,6 @@

subsubsection {* Closure-free presentation *}

-(*Badly needed: a linear arithmetic procedure for multisets*)
-
-lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
-by (simp add: multiset_eq_conv_count_eq)
-
text {* One direction. *}

lemma mult_implies_one_step:
@@ -628,7 +1163,7 @@
apply (rule_tac x = I in exI)
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
- apply (simp (no_asm_simp) add: union_assoc [symmetric])
+ apply (simp (no_asm_simp) add: add_assoc [symmetric])
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
apply (simp add: diff_union_single_conv)
apply (simp (no_asm_use) add: trans_def)
@@ -649,14 +1184,6 @@
apply (simp (no_asm))
done

-lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
-by (simp add: multiset_eq_conv_count_eq)
-
-lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
-apply (erule size_eq_Suc_imp_elem [THEN exE])
-apply (drule elem_imp_eq_diff_union, auto)
-done
-
lemma one_step_implies_mult_aux:
"trans r ==>
\<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))
@@ -679,13 +1206,13 @@
(I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
prefer 2
apply force
-apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
+apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
apply (erule trancl_trans)
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
-apply (simp add: union_ac)
+apply (simp add: add_ac)
done

lemma one_step_implies_mult:
@@ -699,10 +1226,10 @@
instantiation multiset :: (order) order
begin

-definition less_multiset_def [code del]:
+definition less_multiset_def:
"M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"

-definition le_multiset_def [code del]:
+definition le_multiset_def:
"M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"

lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
@@ -776,7 +1303,7 @@
apply auto
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
-apply (simp add: union_assoc)
+apply (simp add: add_assoc)
done

lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
@@ -787,8 +1314,8 @@
done

lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
-apply (subst union_commute [of B C])
-apply (subst union_commute [of D C])
+apply (subst add_commute [of B C])
+apply (subst add_commute [of D C])
apply (erule union_less_mono2)
done

@@ -819,7 +1346,7 @@
qed

lemma union_upper2: "B <= A + (B::'a::order multiset)"
-by (subst union_commute) (rule union_upper1)
+by (subst add_commute) (rule union_upper1)

instance multiset :: (order) pordered_ab_semigroup_add
apply intro_classes
@@ -827,416 +1354,6 @@
done

-subsection {* Link with lists *}
-
-primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
-  "multiset_of [] = {#}" |
-  "multiset_of (a # x) = multiset_of x + {# a #}"
-
-lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
-by (induct x) auto
-
-lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
-by (induct x) auto
-
-lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
-by (induct x) auto
-
-lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
-by (induct xs) auto
-
-lemma multiset_of_append [simp]:
-  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
-by (induct xs arbitrary: ys) (auto simp: union_ac)
-
-lemma surj_multiset_of: "surj multiset_of"
-apply (unfold surj_def)
-apply (rule allI)
-apply (rule_tac M = y in multiset_induct)
- apply auto
-apply (rule_tac x = "x # xa" in exI)
-apply auto
-done
-
-lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
-by (induct x) auto
-
-lemma distinct_count_atmost_1:
-  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
-apply (induct x, simp, rule iffI, simp_all)
-apply (rule conjI)
-apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
-apply (erule_tac x = a in allE, simp, clarify)
-apply (erule_tac x = aa in allE, simp)
-done
-
-lemma multiset_of_eq_setD:
-  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
-by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
-
-lemma set_eq_iff_multiset_of_eq_distinct:
-  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
-    (set x = set y) = (multiset_of x = multiset_of y)"
-by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
-
-lemma set_eq_iff_multiset_of_remdups_eq:
-   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
-apply (rule iffI)
-apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
-apply (drule distinct_remdups [THEN distinct_remdups
-      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
-apply simp
-done
-
-lemma multiset_of_compl_union [simp]:
-  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
-by (induct xs) (auto simp: union_ac)
-
-lemma count_filter:
-  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
-by (induct xs) auto
-
-lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
-apply (induct ls arbitrary: i)
- apply simp
-apply (case_tac i)
- apply auto
-done
-
-lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
-by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
-
-lemma multiset_of_eq_length:
-assumes "multiset_of xs = multiset_of ys"
-shows "length xs = length ys"
-using assms
-proof (induct arbitrary: ys rule: length_induct)
-  case (1 xs ys)
-  show ?case
-  proof (cases xs)
-    case Nil with "1.prems" show ?thesis by simp
-  next
-    case (Cons x xs')
-    note xCons = Cons
-    show ?thesis
-    proof (cases ys)
-      case Nil
-      with "1.prems" Cons show ?thesis by simp
-    next
-      case (Cons y ys')
-      have x_in_ys: "x = y \<or> x \<in> set ys'"
-      proof (cases "x = y")
-        case True then show ?thesis ..
-      next
-        case False
-        from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
-        with False show ?thesis by (simp add: mem_set_multiset_eq)
-      qed
-      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
-        (\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
-      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
-        apply -
-        apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
-        apply fastsimp
-        done
-      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
-      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
-      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
-    qed
-  qed
-qed
-
-text {*
-  This lemma shows which properties suffice to show that a function
-  @{text "f"} with @{text "f xs = ys"} behaves like sort.
-*}
-lemma properties_for_sort:
-  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
-proof (induct xs arbitrary: ys)
-  case Nil then show ?case by simp
-next
-  case (Cons x xs)
-  then have "x \<in> set ys"
-    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
-  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
-    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
-qed
-
-
-subsection {* Pointwise ordering induced by count *}
-
-definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
-  [code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
-
-definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
-  [code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
-
-notation mset_le  (infix "\<subseteq>#" 50)
-notation mset_less  (infix "\<subset>#" 50)
-
-lemma mset_le_refl[simp]: "A \<le># A"
-unfolding mset_le_def by auto
-
-lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
-unfolding mset_le_def by (fast intro: order_trans)
-
-lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
-apply (unfold mset_le_def)
-apply (rule multiset_eq_conv_count_eq [THEN iffD2])
-apply (blast intro: order_antisym)
-done
-
-lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
-apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
-apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
-done
-
-lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
-unfolding mset_le_def by auto
-
-lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
-unfolding mset_le_def by auto
-
-lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
-apply (unfold mset_le_def)
-apply auto
-apply (erule_tac x = a in allE)+
-apply auto
-done
-
-lemma mset_le_add_left[simp]: "A \<le># A + B"
-unfolding mset_le_def by auto
-
-lemma mset_le_add_right[simp]: "B \<le># A + B"
-unfolding mset_le_def by auto
-
-lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
-by (simp add: mset_le_def)
-
-lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
-by (simp add: multiset_eq_conv_count_eq mset_le_def)
-
-lemma mset_le_multiset_union_diff_commute:
-assumes "B \<le># A"
-shows "A - B + C = A + C - B"
-proof -
-  from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
-  from this obtain D where "A = B + D" ..
-  then show ?thesis
-    apply simp
-    apply (subst union_commute)
-    apply (subst multiset_diff_union_assoc)
-    apply simp
-    apply (simp add: diff_cancel)
-    apply (subst union_assoc)
-    apply (subst union_commute[of "B" _])
-    apply (subst multiset_diff_union_assoc)
-    apply simp
-    apply (simp add: diff_cancel)
-    done
-qed
-
-lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
-apply (induct xs)
- apply auto
-apply (rule mset_le_trans)
- apply auto
-done
-
-lemma multiset_of_update:
-  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
-proof (induct ls arbitrary: i)
-  case Nil then show ?case by simp
-next
-  case (Cons x xs)
-  show ?case
-  proof (cases i)
-    case 0 then show ?thesis by simp
-  next
-    case (Suc i')
-    with Cons show ?thesis
-      apply simp
-      apply (subst union_assoc)
-      apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
-      apply (subst union_assoc [symmetric])
-      apply simp
-      apply (rule mset_le_multiset_union_diff_commute)
-      apply (simp add: mset_le_single nth_mem_multiset_of)
-      done
-  qed
-qed
-
-lemma multiset_of_swap:
-  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
-    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
-apply (case_tac "i = j")
- apply simp
-apply (simp add: multiset_of_update)
-apply (subst elem_imp_eq_diff_union[symmetric])
- apply (simp add: nth_mem_multiset_of)
-apply simp
-done
-
-interpretation mset_order: order "op \<le>#" "op <#"
-proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
-  mset_le_trans simp: mset_less_def)
-
-interpretation mset_order_cancel_semigroup:
-  pordered_cancel_ab_semigroup_add "op +" "op \<le>#" "op <#"
-proof qed (erule mset_le_mono_add [OF mset_le_refl])
-
-interpretation mset_order_semigroup_cancel:
-  pordered_ab_semigroup_add_imp_le "op +" "op \<le>#" "op <#"
-proof qed simp
-
-
-lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
-apply (clarsimp simp: mset_le_def mset_less_def)
-apply (erule_tac x=x in allE)
-apply auto
-done
-
-lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
-apply (clarsimp simp: mset_le_def mset_less_def)
-apply (erule_tac x = x in allE)
-apply auto
-done
-
-lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
-apply (rule conjI)
- apply (simp add: mset_lessD)
-apply (clarsimp simp: mset_le_def mset_less_def)
-apply safe
- apply (erule_tac x = a in allE)
- apply (auto split: split_if_asm)
-done
-
-lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
-apply (rule conjI)
- apply (simp add: mset_leD)
-apply (force simp: mset_le_def mset_less_def split: split_if_asm)
-done
-
-lemma mset_less_of_empty[simp]: "A \<subset># {#} = False"
-by (induct A) (auto simp: mset_le_def mset_less_def)
-
-lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
-by (auto simp: mset_le_def mset_less_def)
-
-lemma multi_psub_self[simp]: "A \<subset># A = False"
-by (auto simp: mset_le_def mset_less_def)
-
-lemma mset_less_add_bothsides:
-  "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
-by (auto simp: mset_le_def mset_less_def)
-
-lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
-by (auto simp: mset_le_def mset_less_def)
-
-lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
-proof (induct A arbitrary: B)
-  case (empty M)
-  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
-  then obtain M' x where "M = M' + {#x#}"
-    by (blast dest: multi_nonempty_split)
-  then show ?case by simp
-next
-  case (add S x T)
-  have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
-  have SxsubT: "S + {#x#} \<subset># T" by fact
-  then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
-  then obtain T' where T: "T = T' + {#x#}"
-    by (blast dest: multi_member_split)
-  then have "S \<subset># T'" using SxsubT
-    by (blast intro: mset_less_add_bothsides)
-  then have "size S < size T'" using IH by simp
-  then show ?case using T by simp
-qed
-
-lemmas mset_less_trans = mset_order.less_trans
-
-lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
-by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
-
-
-subsection {* Strong induction and subset induction for multisets *}
-
-text {* Well-foundedness of proper subset operator: *}
-
-text {* proper multiset subset *}
-definition
-  mset_less_rel :: "('a multiset * 'a multiset) set" where
-  "mset_less_rel = {(A,B). A \<subset># B}"
-
-lemma multiset_add_sub_el_shuffle:
-  assumes "c \<in># B" and "b \<noteq> c"
-  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
-proof -
-  from `c \<in># B` obtain A where B: "B = A + {#c#}"
-    by (blast dest: multi_member_split)
-  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
-  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}"
-    by (simp add: union_ac)
-  then show ?thesis using B by simp
-qed
-
-lemma wf_mset_less_rel: "wf mset_less_rel"
-apply (unfold mset_less_rel_def)
-apply (rule wf_measure [THEN wf_subset, where f1=size])
-apply (clarsimp simp: measure_def inv_image_def mset_less_size)
-done
-
-text {* The induction rules: *}
-
-lemma full_multiset_induct [case_names less]:
-assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
-shows "P B"
-apply (rule wf_mset_less_rel [THEN wf_induct])
-apply (rule ih, auto simp: mset_less_rel_def)
-done
-
-lemma multi_subset_induct [consumes 2, case_names empty add]:
-assumes "F \<subseteq># A"
-  and empty: "P {#}"
-  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
-shows "P F"
-proof -
-  from `F \<subseteq># A`
-  show ?thesis
-  proof (induct F)
-    show "P {#}" by fact
-  next
-    fix x F
-    assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
-    show "P (F + {#x#})"
-    proof (rule insert)
-      from i show "x \<in># A" by (auto dest: mset_le_insertD)
-      from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
-      with P show "P F" .
-    qed
-  qed
-qed
-
-text{* A consequence: Extensionality. *}
-
-lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
-apply (rule iffI)
- prefer 2
- apply clarsimp
-apply (induct A arbitrary: B rule: full_multiset_induct)
-apply (rename_tac C)
-apply (case_tac B rule: multiset_cases)
- apply (simp add: empty_multiset_count)
-apply simp
-apply (case_tac "x \<in># C")
- apply (force dest: multi_member_split)
-apply (erule_tac x = x in allE)
-apply simp
-done
-
-lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
-
-
subsection {* The fold combinator *}

text {*
@@ -1282,9 +1399,7 @@
lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
unfolding fold_mset_def by blast

-locale left_commutative =
-fixes f :: "'a => 'b => 'b"
-assumes left_commute: "f x (f y z) = f y (f x z)"
+context fun_left_comm
begin

lemma fold_msetG_determ:
@@ -1324,7 +1439,7 @@
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
by (auto intro: insert_noteq_member dest: sym)
have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
-        then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
+        then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_order.less_trans)
from MBb MCc have "B + {#b#} = C + {#c#}" by blast
then have [simp]: "B + {#b#} - {#c#} = C"
using MBb MCc binC cinB by auto
@@ -1342,7 +1457,7 @@
dest: fold_msetG.insertI [where x=b])
then have "f b d = v" using IH CsubM Cv by blast
ultimately show ?thesis using x\<^isub>1 x\<^isub>2
-          by (auto simp: left_commute)
+          by (auto simp: fun_left_comm)
qed
qed
qed
@@ -1363,7 +1478,7 @@

lemma fold_mset_insert:
"fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
-apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)
+apply (simp add: fold_mset_def fold_mset_insert_aux add_commute)
apply (rule the_equality)
apply (auto cong add: conj_cong
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
@@ -1378,7 +1493,7 @@
done

lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
-by (induct A) (auto simp: fold_mset_insert left_commute [of x])
+by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x])

lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
using fold_mset_insert [of z "{#}"] by simp
@@ -1389,7 +1504,7 @@
case empty then show ?case by simp
next
case (add A x)
-  have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
+  have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac)
then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
by (simp add: fold_mset_insert)
also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
@@ -1398,10 +1513,10 @@
qed

lemma fold_mset_fusion:
-  assumes "left_commutative g"
+  assumes "fun_left_comm g"
shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
proof -
-  interpret left_commutative g by fact
+  interpret fun_left_comm g by (fact assms)
show "PROP ?P" by (induct A) auto
qed

@@ -1430,11 +1545,11 @@

subsection {* Image *}

-definition [code del]:
- "image_mset f = fold_mset (op + o single o f) {#}"
+definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
+  "image_mset f = fold_mset (op + o single o f) {#}"

-interpretation image_left_comm: left_commutative "op + o single o f"
-  proof qed (simp add:union_ac)
+interpretation image_left_comm: fun_left_comm "op + o single o f"
+proof qed (simp add: add_ac)

lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
by (simp add: image_mset_def)
@@ -1450,7 +1565,7 @@
"image_mset f (M+N) = image_mset f M + image_mset f N"
apply (induct N)
apply simp
-apply (simp add: union_assoc [symmetric] image_mset_insert)
+apply (simp add: add_assoc [symmetric] image_mset_insert)
done

lemma size_image_mset [simp]: "size (image_mset f M) = size M"
@@ -1608,7 +1723,7 @@

val regroup_munion_conv =
Function_Lib.regroup_conv @{const_name Multiset.Mempty} @{const_name plus}
-        (map (fn t => t RS eq_reflection) (@{thms union_ac} @ @{thms empty_idemp}))
+        (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_idemp}))

fun unfold_pwleq_tac i =
(rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
@@ -1629,4 +1744,31 @@
end
*}

-end
+
+subsection {* Legacy theorem bindings *}
+
+lemmas multi_count_eq = multiset_eq_conv_count_eq [symmetric]
+
+lemma union_commute: "M + N = N + (M::'a multiset)"
+  by (fact add_commute)
+
+lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
+  by (fact add_assoc)
+
+lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
+  by (fact add_left_commute)
+
+lemmas union_ac = union_assoc union_commute union_lcomm
+
+lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"
+  by (fact add_right_cancel)
+
+lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"
+  by (fact add_left_cancel)
+
+lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
+  by (fact add_imp_eq)
+
+lemmas mset_less_trans = mset_order.less_trans
+
+end
\ No newline at end of file```