src/HOL/Library/Multiset.thy

syntax for multiset membership modelled after syntax for set membership

(*  Title:      HOL/Library/Multiset.thy
    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
    Author:     Andrei Popescu, TU Muenchen
    Author:     Jasmin Blanchette, Inria, LORIA, MPII
    Author:     Dmitriy Traytel, TU Muenchen
    Author:     Mathias Fleury, MPII
*)

section \<open>(Finite) multisets\<close>

theory Multiset
imports Main
begin

subsection \<open>The type of multisets\<close>

definition "multiset = {f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}"

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

setup_lifting type_definition_multiset

lemma multiset_eq_iff: "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"
  by (simp only: count_inject [symmetric] fun_eq_iff)

lemma multiset_eqI: "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"
  using multiset_eq_iff by auto

text \<open>Preservation of the representing set @{term multiset}.\<close>

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 filter_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 filter_preserves_multiset


subsection \<open>Representing multisets\<close>

text \<open>Multiset enumeration\<close>

instantiation multiset :: (type) cancel_comm_monoid_add
begin

lift_definition zero_multiset :: "'a multiset" is "\<lambda>a. 0"
by (rule const0_in_multiset)

abbreviation Mempty :: "'a multiset" ("{#}") where
  "Mempty \<equiv> 0"

lift_definition plus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda>M N. (\<lambda>a. M a + N a)"
by (rule union_preserves_multiset)

lift_definition minus_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is "\<lambda> M N. \<lambda>a. M a - N a"
by (rule diff_preserves_multiset)

instance
  by (standard; transfer; simp add: fun_eq_iff)

end

lift_definition single :: "'a \<Rightarrow> 'a multiset" is "\<lambda>a b. if b = a then 1 else 0"
by (rule only1_in_multiset)

syntax
  "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
translations
  "{#x, xs#}" == "{#x#} + {#xs#}"
  "{#x#}" == "CONST single x"

lemma count_empty [simp]: "count {#} a = 0"
  by (simp add: zero_multiset.rep_eq)

lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
  by (simp add: single.rep_eq)


subsection \<open>Basic operations\<close>

subsubsection \<open>Conversion to set and membership\<close>

definition set_mset :: "'a multiset \<Rightarrow> 'a set"
  where "set_mset M = {x. count M x > 0}"

abbreviation Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool"
  where "Melem a M \<equiv> a \<in> set_mset M"

notation
  Melem  ("op \<in>#") and
  Melem  ("(_/ \<in># _)" [51, 51] 50)

notation  (ASCII)
  Melem  ("op :#") and
  Melem  ("(_/ :# _)" [51, 51] 50)

abbreviation not_Melem :: "'a \<Rightarrow> 'a multiset \<Rightarrow> bool"
  where "not_Melem a M \<equiv> a \<notin> set_mset M"

notation
  not_Melem  ("op \<notin>#") and
  not_Melem  ("(_/ \<notin># _)" [51, 51] 50)

notation  (ASCII)
  not_Melem  ("op ~:#") and
  not_Melem  ("(_/ ~:# _)" [51, 51] 50)

context
begin

qualified abbreviation Ball :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
  where "Ball M \<equiv> Set.Ball (set_mset M)"

qualified abbreviation Bex :: "'a multiset \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
  where "Bex M \<equiv> Set.Bex (set_mset M)"

end

syntax
  "_MBall"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>_\<in>#_./ _)" [0, 0, 10] 10)
  "_MBex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>_\<in>#_./ _)" [0, 0, 10] 10)

syntax  (ASCII)
  "_MBall"       :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<forall>_:#_./ _)" [0, 0, 10] 10)
  "_MBex"        :: "pttrn \<Rightarrow> 'a set \<Rightarrow> bool \<Rightarrow> bool"      ("(3\<exists>_:#_./ _)" [0, 0, 10] 10)

translations
  "\<forall>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Ball A (\<lambda>x. P)"
  "\<exists>x\<in>#A. P" \<rightleftharpoons> "CONST Multiset.Bex A (\<lambda>x. P)"

lemma count_eq_zero_iff:
  "count M x = 0 \<longleftrightarrow> x \<notin># M"
  by (auto simp add: set_mset_def)

lemma not_in_iff:
  "x \<notin># M \<longleftrightarrow> count M x = 0"
  by (auto simp add: count_eq_zero_iff)

lemma count_greater_zero_iff [simp]:
  "count M x > 0 \<longleftrightarrow> x \<in># M"
  by (auto simp add: set_mset_def)

lemma count_inI:
  assumes "count M x = 0 \<Longrightarrow> False"
  shows "x \<in># M"
proof (rule ccontr)
  assume "x \<notin># M"
  with assms show False by (simp add: not_in_iff)
qed

lemma in_countE:
  assumes "x \<in># M"
  obtains n where "count M x = Suc n"
proof -
  from assms have "count M x > 0" by simp
  then obtain n where "count M x = Suc n"
    using gr0_conv_Suc by blast
  with that show thesis .
qed

lemma count_greater_eq_Suc_zero_iff [simp]:
  "count M x \<ge> Suc 0 \<longleftrightarrow> x \<in># M"
  by (simp add: Suc_le_eq)

lemma count_greater_eq_one_iff [simp]:
  "count M x \<ge> 1 \<longleftrightarrow> x \<in># M"
  by simp

lemma set_mset_empty [simp]:
  "set_mset {#} = {}"
  by (simp add: set_mset_def)

lemma set_mset_single [simp]:
  "set_mset {#b#} = {b}"
  by (simp add: set_mset_def)

lemma set_mset_eq_empty_iff [simp]:
  "set_mset M = {} \<longleftrightarrow> M = {#}"
  by (auto simp add: multiset_eq_iff count_eq_zero_iff)

lemma finite_set_mset [iff]:
  "finite (set_mset M)"
  using count [of M] by (simp add: multiset_def)


subsubsection \<open>Union\<close>

lemma count_union [simp]:
  "count (M + N) a = count M a + count N a"
  by (simp add: plus_multiset.rep_eq)

lemma set_mset_union [simp]:
  "set_mset (M + N) = set_mset M \<union> set_mset N"
  by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_union) simp


subsubsection \<open>Difference\<close>

instance multiset :: (type) comm_monoid_diff
  by standard (transfer; simp add: fun_eq_iff)

lemma count_diff [simp]:
  "count (M - N) a = count M a - count N a"
  by (simp add: minus_multiset.rep_eq)

lemma in_diff_count:
  "a \<in># M - N \<longleftrightarrow> count N a < count M a"
  by (simp add: set_mset_def)

lemma count_in_diffI:
  assumes "\<And>n. count N x = n + count M x \<Longrightarrow> False"
  shows "x \<in># M - N"
proof (rule ccontr)
  assume "x \<notin># M - N"
  then have "count N x = (count N x - count M x) + count M x"
    by (simp add: in_diff_count not_less)
  with assms show False by auto
qed

lemma in_diff_countE:
  assumes "x \<in># M - N"
  obtains n where "count M x = Suc n + count N x"
proof -
  from assms have "count M x - count N x > 0" by (simp add: in_diff_count)
  then have "count M x > count N x" by simp
  then obtain n where "count M x = Suc n + count N x"
    using less_iff_Suc_add by auto
  with that show thesis .
qed

lemma in_diffD:
  assumes "a \<in># M - N"
  shows "a \<in># M"
proof -
  have "0 \<le> count N a" by simp
  also from assms have "count N a < count M a"
    by (simp add: in_diff_count)
  finally show ?thesis by simp
qed

lemma set_mset_diff:
  "set_mset (M - N) = {a. count N a < count M a}"
  by (simp add: set_mset_def)

lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
  by rule (fact Groups.diff_zero, fact Groups.zero_diff)

lemma diff_cancel [simp]: "A - A = {#}"
  by (fact Groups.diff_cancel)

lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"
  by (fact add_diff_cancel_right')

lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"
  by (fact add_diff_cancel_left')

lemma diff_right_commute:
  fixes M N Q :: "'a multiset"
  shows "M - N - Q = M - Q - N"
  by (fact diff_right_commute)

lemma diff_add:
  fixes M N Q :: "'a multiset"
  shows "M - (N + Q) = M - N - Q"
  by (rule sym) (fact diff_diff_add)

lemma insert_DiffM: "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
  by (clarsimp simp: multiset_eq_iff)

lemma insert_DiffM2 [simp]: "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
  by (clarsimp simp: multiset_eq_iff)

lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"
  by (auto simp add: multiset_eq_iff)

lemma diff_union_single_conv:
  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"
  by (simp add: multiset_eq_iff Suc_le_eq)

lemma mset_add [elim?]:
  assumes "a \<in># A"
  obtains B where "A = B + {#a#}"
proof -
  from assms have "A = (A - {#a#}) + {#a#}"
    by simp
  with that show thesis .
qed

lemma union_iff:
  "a \<in># A + B \<longleftrightarrow> a \<in># A \<or> a \<in># B"
  by auto


subsubsection \<open>Equality of multisets\<close>

lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
  by (simp add: multiset_eq_iff)

lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"
  by (auto simp add: multiset_eq_iff)

lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"
  by (auto simp add: multiset_eq_iff)

lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"
  by (auto simp add: multiset_eq_iff)

lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"
  by (auto simp add: multiset_eq_iff)

lemma diff_single_trivial: "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"
  by (auto simp add: multiset_eq_iff not_in_iff)

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
  show ?lhs if ?rhs using that by auto
  show ?rhs if ?lhs
    by (metis Multiset.diff_cancel add.commute add_diff_cancel_left' diff_add_zero diff_single_trivial insert_DiffM that)
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 \<longleftrightarrow> ?rhs")
(* shorter: by (simp add: multiset_eq_iff) fastforce *)
proof
  show ?lhs if ?rhs
    using that
    by (auto simp add: add.assoc add.commute [of "{#b#}"])
      (drule sym, simp add: add.assoc [symmetric])
  show ?rhs if ?lhs
  proof (cases "a = b")
    case True with \<open>?lhs\<close> show ?thesis by simp
  next
    case False
    from \<open>?lhs\<close> have "a \<in># N + {#b#}" by (rule union_single_eq_member)
    with False have "a \<in># N" by auto
    moreover from \<open>?lhs\<close> 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)

lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
  by (rule exI [where x = "M - {#x#}"]) simp

lemma multiset_add_sub_el_shuffle:
  assumes "c \<in># B"
    and "b \<noteq> c"
  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
proof -
  from \<open>c \<in># B\<close> 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: ac_simps)
  then show ?thesis using B by simp
qed


subsubsection \<open>Pointwise ordering induced by count\<close>

definition subseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<subseteq>#" 50)
  where "A \<subseteq># B = (\<forall>a. count A a \<le> count B a)"

definition subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<subset>#" 50)
  where "A \<subset># B = (A \<subseteq># B \<and> A \<noteq> B)"

abbreviation (input) supseteq_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<supseteq>#" 50)
  where "supseteq_mset A B \<equiv> B \<subseteq># A"

abbreviation (input) supset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<supset>#" 50)
  where "supset_mset A B \<equiv> B \<subset># A"

notation (input)
  subseteq_mset  (infix "\<le>#" 50) and
  supseteq_mset  (infix "\<ge>#" 50)

notation (ASCII)
  subseteq_mset  (infix "<=#" 50) and
  subset_mset  (infix "<#" 50) and
  supseteq_mset  (infix ">=#" 50) and
  supset_mset  (infix ">#" 50)

interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op \<subseteq>#" "op \<subset>#"
  by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
  -- \<open>FIXME: avoid junk stemming from type class interpretation\<close>

lemma mset_less_eqI:
  "(\<And>a. count A a \<le> count B a) \<Longrightarrow> A \<subseteq># B"
  by (simp add: subseteq_mset_def)

lemma mset_less_eq_count:
  "A \<subseteq># B \<Longrightarrow> count A a \<le> count B a"
  by (simp add: subseteq_mset_def)

lemma mset_le_exists_conv: "(A::'a multiset) \<subseteq># B \<longleftrightarrow> (\<exists>C. B = A + C)"
  unfolding subseteq_mset_def
  apply (rule iffI)
   apply (rule exI [where x = "B - A"])
   apply (auto intro: multiset_eq_iff [THEN iffD2])
  done

interpretation subset_mset: ordered_cancel_comm_monoid_diff  "op +" 0 "op \<le>#" "op <#" "op -"
  by standard (simp, fact mset_le_exists_conv)

declare subset_mset.zero_order[simp del]
  -- \<open>this removes some simp rules not in the usual order for multisets\<close>

lemma mset_le_mono_add_right_cancel [simp]: "(A::'a multiset) + C \<subseteq># B + C \<longleftrightarrow> A \<subseteq># B"
   by (fact subset_mset.add_le_cancel_right)
 
lemma mset_le_mono_add_left_cancel [simp]: "C + (A::'a multiset) \<subseteq># C + B \<longleftrightarrow> A \<subseteq># B"
   by (fact subset_mset.add_le_cancel_left)
 
lemma mset_le_mono_add: "(A::'a multiset) \<subseteq># B \<Longrightarrow> C \<subseteq># D \<Longrightarrow> A + C \<subseteq># B + D"
   by (fact subset_mset.add_mono)
 
lemma mset_le_add_left [simp]: "(A::'a multiset) \<subseteq># A + B"
   unfolding subseteq_mset_def by auto
 
lemma mset_le_add_right [simp]: "B \<subseteq># (A::'a multiset) + B"
   unfolding subseteq_mset_def by auto
 
lemma single_subset_iff [simp]:
  "{#a#} \<subseteq># M \<longleftrightarrow> a \<in># M"
  by (auto simp add: subseteq_mset_def Suc_le_eq)

lemma mset_le_single: "a \<in># B \<Longrightarrow> {#a#} \<subseteq># B"
  by (simp add: subseteq_mset_def Suc_le_eq)
 
lemma multiset_diff_union_assoc:
  fixes A B C D :: "'a multiset"
  shows "C \<subseteq># B \<Longrightarrow> A + B - C = A + (B - C)"
  by (fact subset_mset.diff_add_assoc)
 
lemma mset_le_multiset_union_diff_commute:
  fixes A B C D :: "'a multiset"
  shows "B \<subseteq># A \<Longrightarrow> A - B + C = A + C - B"
  by (fact subset_mset.add_diff_assoc2)

lemma diff_le_self[simp]:
  "(M::'a multiset) - N \<subseteq># M"
  by (simp add: subseteq_mset_def)

lemma mset_leD:
  assumes "A \<subseteq># B" and "x \<in># A"
  shows "x \<in># B"
proof -
  from \<open>x \<in># A\<close> have "count A x > 0" by simp
  also from \<open>A \<subseteq># B\<close> have "count A x \<le> count B x"
    by (simp add: subseteq_mset_def)
  finally show ?thesis by simp
qed
  
lemma mset_lessD:
  "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
  by (auto intro: mset_leD [of A])

lemma set_mset_mono:
  "A \<subseteq># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
  by (metis mset_leD subsetI)

lemma mset_le_insertD:
  "A + {#x#} \<subseteq># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
apply (rule conjI)
 apply (simp add: mset_leD)
 apply (clarsimp simp: subset_mset_def subseteq_mset_def)
 apply safe
  apply (erule_tac x = a in allE)
  apply (auto split: if_split_asm)
done

lemma mset_less_insertD:
  "A + {#x#} \<subset># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
  by (rule mset_le_insertD) simp

lemma mset_less_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False"
  by (auto simp add: subseteq_mset_def subset_mset_def multiset_eq_iff)

lemma empty_le [simp]: "{#} \<subseteq># A"
  unfolding mset_le_exists_conv by auto
 
lemma insert_subset_eq_iff:
  "{#a#} + A \<subseteq># B \<longleftrightarrow> a \<in># B \<and> A \<subseteq># B - {#a#}"
  using le_diff_conv2 [of "Suc 0" "count B a" "count A a"]
  apply (auto simp add: subseteq_mset_def not_in_iff Suc_le_eq)
  apply (rule ccontr)
  apply (auto simp add: not_in_iff)
  done

lemma insert_union_subset_iff:
  "{#a#} + A \<subset># B \<longleftrightarrow> a \<in># B \<and> A \<subset># B - {#a#}"
  by (auto simp add: insert_subset_eq_iff subset_mset_def insert_DiffM)

lemma subset_eq_diff_conv:
  "A - C \<subseteq># B \<longleftrightarrow> A \<subseteq># B + C"
  by (simp add: subseteq_mset_def le_diff_conv)

lemma le_empty [simp]: "M \<subseteq># {#} \<longleftrightarrow> M = {#}"
  unfolding mset_le_exists_conv by auto

lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
  by (auto simp: subset_mset_def subseteq_mset_def)

lemma multi_psub_self[simp]: "(A::'a multiset) \<subset># A = False"
  by simp

lemma mset_less_add_bothsides: "N + {#x#} \<subset># M + {#x#} \<Longrightarrow> N \<subset># M"
  by (fact subset_mset.add_less_imp_less_right)

lemma mset_less_empty_nonempty: "{#} \<subset># S \<longleftrightarrow> S \<noteq> {#}"
  by (fact subset_mset.zero_less_iff_neq_zero)

lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
  by (auto simp: subset_mset_def elim: mset_add)


subsubsection \<open>Intersection\<close>

definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
  multiset_inter_def: "inf_subset_mset A B = A - (A - B)"

interpretation subset_mset: semilattice_inf inf_subset_mset "op \<subseteq>#" "op \<subset>#"
proof -
  have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat
    by arith
  show "class.semilattice_inf op #\<inter> op \<subseteq># op \<subset>#"
    by standard (auto simp add: multiset_inter_def subseteq_mset_def)
qed
  -- \<open>FIXME: avoid junk stemming from type class interpretation\<close>

lemma multiset_inter_count [simp]:
  fixes A B :: "'a multiset"
  shows "count (A #\<inter> B) x = min (count A x) (count B x)"
  by (simp add: multiset_inter_def)

lemma set_mset_inter [simp]:
  "set_mset (A #\<inter> B) = set_mset A \<inter> set_mset B"
  by (simp only: set_eq_iff count_greater_zero_iff [symmetric] multiset_inter_count) simp

lemma diff_intersect_left_idem [simp]:
  "M - M #\<inter> N = M - N"
  by (simp add: multiset_eq_iff min_def)

lemma diff_intersect_right_idem [simp]:
  "M - N #\<inter> M = M - N"
  by (simp add: multiset_eq_iff min_def)

lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
  by (rule multiset_eqI) auto

lemma multiset_union_diff_commute:
  assumes "B #\<inter> C = {#}"
  shows "A + B - C = A - C + B"
proof (rule multiset_eqI)
  fix x
  from assms have "min (count B x) (count C x) = 0"
    by (auto simp add: multiset_eq_iff)
  then have "count B x = 0 \<or> count C x = 0"
    unfolding min_def by (auto split: if_splits)
  then show "count (A + B - C) x = count (A - C + B) x"
    by auto
qed

lemma disjunct_not_in:
  "A #\<inter> B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)" (is "?P \<longleftrightarrow> ?Q")
proof
  assume ?P
  show ?Q
  proof
    fix a
    from \<open>?P\<close> have "min (count A a) (count B a) = 0"
      by (simp add: multiset_eq_iff)
    then have "count A a = 0 \<or> count B a = 0"
      by (cases "count A a \<le> count B a") (simp_all add: min_def)
    then show "a \<notin># A \<or> a \<notin># B"
      by (simp add: not_in_iff)
  qed
next
  assume ?Q
  show ?P
  proof (rule multiset_eqI)
    fix a
    from \<open>?Q\<close> have "count A a = 0 \<or> count B a = 0"
      by (auto simp add: not_in_iff)
    then show "count (A #\<inter> B) a = count {#} a"
      by auto
  qed
qed

lemma empty_inter [simp]: "{#} #\<inter> M = {#}"
  by (simp add: multiset_eq_iff)

lemma inter_empty [simp]: "M #\<inter> {#} = {#}"
  by (simp add: multiset_eq_iff)

lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = M #\<inter> N"
  by (simp add: multiset_eq_iff not_in_iff)

lemma inter_add_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<inter> N = (M #\<inter> (N - {#x#})) + {#x#}"
  by (auto simp add: multiset_eq_iff elim: mset_add)

lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = N #\<inter> M"
  by (simp add: multiset_eq_iff not_in_iff)

lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (M + {#x#}) = ((N - {#x#}) #\<inter> M) + {#x#}"
  by (auto simp add: multiset_eq_iff elim: mset_add)

lemma disjunct_set_mset_diff:
  assumes "M #\<inter> N = {#}"
  shows "set_mset (M - N) = set_mset M"
proof (rule set_eqI)
  fix a
  from assms have "a \<notin># M \<or> a \<notin># N"
    by (simp add: disjunct_not_in)
  then show "a \<in># M - N \<longleftrightarrow> a \<in># M"
    by (auto dest: in_diffD) (simp add: in_diff_count not_in_iff)
qed

lemma at_most_one_mset_mset_diff:
  assumes "a \<notin># M - {#a#}"
  shows "set_mset (M - {#a#}) = set_mset M - {a}"
  using assms by (auto simp add: not_in_iff in_diff_count set_eq_iff)

lemma more_than_one_mset_mset_diff:
  assumes "a \<in># M - {#a#}"
  shows "set_mset (M - {#a#}) = set_mset M"
proof (rule set_eqI)
  fix b
  have "Suc 0 < count M b \<Longrightarrow> count M b > 0" by arith
  then show "b \<in># M - {#a#} \<longleftrightarrow> b \<in># M"
    using assms by (auto simp add: in_diff_count)
qed

lemma inter_iff:
  "a \<in># A #\<inter> B \<longleftrightarrow> a \<in># A \<and> a \<in># B"
  by simp

lemma inter_union_distrib_left:
  "A #\<inter> B + C = (A + C) #\<inter> (B + C)"
  by (simp add: multiset_eq_iff min_add_distrib_left)

lemma inter_union_distrib_right:
  "C + A #\<inter> B = (C + A) #\<inter> (C + B)"
  using inter_union_distrib_left [of A B C] by (simp add: ac_simps)

lemma inter_subset_eq_union:
  "A #\<inter> B \<subseteq># A + B"
  by (auto simp add: subseteq_mset_def)


subsubsection \<open>Bounded union\<close>

definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70)
  where "sup_subset_mset A B = A + (B - A)" -- \<open>FIXME irregular fact name\<close>

interpretation subset_mset: semilattice_sup sup_subset_mset "op \<subseteq>#" "op \<subset>#"
proof -
  have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
    by arith
  show "class.semilattice_sup op #\<union> op \<subseteq># op \<subset>#"
    by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
qed
  -- \<open>FIXME: avoid junk stemming from type class interpretation\<close>

lemma sup_subset_mset_count [simp]: -- \<open>FIXME irregular fact name\<close>
  "count (A #\<union> B) x = max (count A x) (count B x)"
  by (simp add: sup_subset_mset_def)

lemma set_mset_sup [simp]:
  "set_mset (A #\<union> B) = set_mset A \<union> set_mset B"
  by (simp only: set_eq_iff count_greater_zero_iff [symmetric] sup_subset_mset_count)
    (auto simp add: not_in_iff elim: mset_add)

lemma empty_sup [simp]: "{#} #\<union> M = M"
  by (simp add: multiset_eq_iff)

lemma sup_empty [simp]: "M #\<union> {#} = M"
  by (simp add: multiset_eq_iff)

lemma sup_union_left1: "\<not> x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> N) + {#x#}"
  by (simp add: multiset_eq_iff not_in_iff)

lemma sup_union_left2: "x \<in># N \<Longrightarrow> (M + {#x#}) #\<union> N = (M #\<union> (N - {#x#})) + {#x#}"
  by (simp add: multiset_eq_iff)

lemma sup_union_right1: "\<not> x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = (N #\<union> M) + {#x#}"
  by (simp add: multiset_eq_iff not_in_iff)

lemma sup_union_right2: "x \<in># N \<Longrightarrow> N #\<union> (M + {#x#}) = ((N - {#x#}) #\<union> M) + {#x#}"
  by (simp add: multiset_eq_iff)

lemma sup_union_distrib_left:
  "A #\<union> B + C = (A + C) #\<union> (B + C)"
  by (simp add: multiset_eq_iff max_add_distrib_left)

lemma union_sup_distrib_right:
  "C + A #\<union> B = (C + A) #\<union> (C + B)"
  using sup_union_distrib_left [of A B C] by (simp add: ac_simps)

lemma union_diff_inter_eq_sup:
  "A + B - A #\<inter> B = A #\<union> B"
  by (auto simp add: multiset_eq_iff)

lemma union_diff_sup_eq_inter:
  "A + B - A #\<union> B = A #\<inter> B"
  by (auto simp add: multiset_eq_iff)


subsubsection \<open>Subset is an order\<close>

interpretation subset_mset: order "op \<le>#" "op <#" by unfold_locales auto


subsubsection \<open>Filter (with comprehension syntax)\<close>

text \<open>Multiset comprehension\<close>

lift_definition filter_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"
is "\<lambda>P M. \<lambda>x. if P x then M x else 0"
by (rule filter_preserves_multiset)

syntax (ASCII)
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")
syntax
  "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")
translations
  "{#x \<in># M. P#}" == "CONST filter_mset (\<lambda>x. P) M"

lemma count_filter_mset [simp]:
  "count (filter_mset P M) a = (if P a then count M a else 0)"
  by (simp add: filter_mset.rep_eq)

lemma set_mset_filter [simp]:
  "set_mset (filter_mset P M) = {a \<in> set_mset M. P a}"
  by (simp only: set_eq_iff count_greater_zero_iff [symmetric] count_filter_mset) simp

lemma filter_empty_mset [simp]: "filter_mset P {#} = {#}"
  by (rule multiset_eqI) simp

lemma filter_single_mset [simp]: "filter_mset P {#x#} = (if P x then {#x#} else {#})"
  by (rule multiset_eqI) simp

lemma filter_union_mset [simp]: "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
  by (rule multiset_eqI) simp

lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
  by (rule multiset_eqI) simp

lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
  by (rule multiset_eqI) simp

lemma multiset_filter_subset[simp]: "filter_mset f M \<subseteq># M"
  by (simp add: mset_less_eqI)

lemma multiset_filter_mono:
  assumes "A \<subseteq># B"
  shows "filter_mset f A \<subseteq># filter_mset f B"
proof -
  from assms[unfolded mset_le_exists_conv]
  obtain C where B: "B = A + C" by auto
  show ?thesis unfolding B by auto
qed

lemma filter_mset_eq_conv:
  "filter_mset P M = N \<longleftrightarrow> N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a)" (is "?P \<longleftrightarrow> ?Q")
proof
  assume ?P then show ?Q by auto (simp add: multiset_eq_iff in_diff_count)
next
  assume ?Q
  then obtain Q where M: "M = N + Q"
    by (auto simp add: mset_le_exists_conv)
  then have MN: "M - N = Q" by simp
  show ?P
  proof (rule multiset_eqI)
    fix a
    from \<open>?Q\<close> MN have *: "\<not> P a \<Longrightarrow> a \<notin># N" "P a \<Longrightarrow> a \<notin># Q"
      by auto
    show "count (filter_mset P M) a = count N a"
    proof (cases "a \<in># M")
      case True
      with * show ?thesis
        by (simp add: not_in_iff M)
    next
      case False then have "count M a = 0"
        by (simp add: not_in_iff)
      with M show ?thesis by simp
    qed 
  qed
qed


subsubsection \<open>Size\<close>

definition wcount where "wcount f M = (\<lambda>x. count M x * Suc (f x))"

lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"
  by (auto simp: wcount_def add_mult_distrib)

definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
  "size_multiset f M = setsum (wcount f M) (set_mset M)"

lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]

instantiation multiset :: (type) size
begin

definition size_multiset where
  size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (\<lambda>_. 0)"
instance ..

end

lemmas size_multiset_overloaded_eq =
  size_multiset_overloaded_def[THEN fun_cong, unfolded size_multiset_eq, simplified]

lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"
by (simp add: size_multiset_def)

lemma size_empty [simp]: "size {#} = 0"
by (simp add: size_multiset_overloaded_def)

lemma size_multiset_single [simp]: "size_multiset f {#b#} = Suc (f b)"
by (simp add: size_multiset_eq)

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

lemma setsum_wcount_Int:
  "finite A \<Longrightarrow> setsum (wcount f N) (A \<inter> set_mset N) = setsum (wcount f N) A"
  by (induct rule: finite_induct)
    (simp_all add: Int_insert_left wcount_def count_eq_zero_iff)

lemma size_multiset_union [simp]:
  "size_multiset f (M + N::'a multiset) = size_multiset f M + size_multiset f N"
apply (simp add: size_multiset_def setsum_Un_nat setsum.distrib setsum_wcount_Int wcount_union)
apply (subst Int_commute)
apply (simp add: setsum_wcount_Int)
done

lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
by (auto simp add: size_multiset_overloaded_def)

lemma size_multiset_eq_0_iff_empty [iff]:
  "size_multiset f M = 0 \<longleftrightarrow> M = {#}"
  by (auto simp add: size_multiset_eq count_eq_zero_iff)

lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
by (auto simp add: size_multiset_overloaded_def)

lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)

lemma size_eq_Suc_imp_elem: "size M = Suc n \<Longrightarrow> \<exists>a. a \<in># M"
apply (unfold size_multiset_overloaded_eq)
apply (drule setsum_SucD)
apply auto
done

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 size_mset_mono:
  fixes A B :: "'a multiset"
  assumes "A \<subseteq># B"
  shows "size A \<le> size B"
proof -
  from assms[unfolded mset_le_exists_conv]
  obtain C where B: "B = A + C" by auto
  show ?thesis unfolding B by (induct C) auto
qed

lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) \<le> size M"
by (rule size_mset_mono[OF multiset_filter_subset])

lemma size_Diff_submset:
  "M \<subseteq># M' \<Longrightarrow> size (M' - M) = size M' - size(M::'a multiset)"
by (metis add_diff_cancel_left' size_union mset_le_exists_conv)


subsection \<open>Induction and case splits\<close>

theorem multiset_induct [case_names empty add, induct type: multiset]:
  assumes empty: "P {#}"
  assumes add: "\<And>M x. P M \<Longrightarrow> P (M + {#x#})"
  shows "P M"
proof (induct n \<equiv> "size M" arbitrary: M)
  case 0 thus "P M" by (simp add: empty)
next
  case (Suc k)
  obtain N x where "M = N + {#x#}"
    using \<open>Suc k = size M\<close> [symmetric]
    using size_eq_Suc_imp_eq_union by fast
  with Suc add show "P M" by simp
qed

lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
by (induct M) auto

lemma multiset_cases [cases type]:
  obtains (empty) "M = {#}"
    | (add) N x where "M = N + {#x#}"
  using assms by (induct M) simp_all

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\<in>#M. P x #} + {# x\<in>#M. \<not> P x #}"
apply (subst multiset_eq_iff)
apply auto
done

lemma mset_less_size: "(A::'a multiset) \<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

lemma size_1_singleton_mset: "size M = 1 \<Longrightarrow> \<exists>a. M = {#a#}"
by (cases M) auto


subsubsection \<open>Strong induction and subset induction for multisets\<close>

text \<open>Well-foundedness of strict subset relation\<close>

lemma wf_less_mset_rel: "wf {(M, N :: 'a multiset). M \<subset># N}"
apply (rule wf_measure [THEN wf_subset, where f1=size])
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
done

lemma full_multiset_induct [case_names less]:
assumes ih: "\<And>B. \<forall>(A::'a multiset). A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
shows "P B"
apply (rule wf_less_mset_rel [THEN wf_induct])
apply (rule ih, auto)
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 \<open>F \<subseteq># A\<close>
  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 \<open>The fold combinator\<close>

definition fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b"
where
  "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"

lemma fold_mset_empty [simp]: "fold_mset f s {#} = s"
  by (simp add: fold_mset_def)

context comp_fun_commute
begin

lemma fold_mset_insert: "fold_mset f s (M + {#x#}) = f x (fold_mset f s M)"
proof -
  interpret mset: comp_fun_commute "\<lambda>y. f y ^^ count M y"
    by (fact comp_fun_commute_funpow)
  interpret mset_union: comp_fun_commute "\<lambda>y. f y ^^ count (M + {#x#}) y"
    by (fact comp_fun_commute_funpow)
  show ?thesis
  proof (cases "x \<in> set_mset M")
    case False
    then have *: "count (M + {#x#}) x = 1"
      by (simp add: not_in_iff)
    from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s (set_mset M) =
      Finite_Set.fold (\<lambda>y. f y ^^ count M y) s (set_mset M)"
      by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
    with False * show ?thesis
      by (simp add: fold_mset_def del: count_union)
  next
    case True
    def N \<equiv> "set_mset M - {x}"
    from N_def True have *: "set_mset M = insert x N" "x \<notin> N" "finite N" by auto
    then have "Finite_Set.fold (\<lambda>y. f y ^^ count (M + {#x#}) y) s N =
      Finite_Set.fold (\<lambda>y. f y ^^ count M y) s N"
      by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
    with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
  qed
qed

corollary fold_mset_single [simp]: "fold_mset f s {#x#} = f x s"
proof -
  have "fold_mset f s ({#} + {#x#}) = f x s" by (simp only: fold_mset_insert) simp
  then show ?thesis by simp
qed

lemma fold_mset_fun_left_comm: "f x (fold_mset f s M) = fold_mset f (f x s) M"
  by (induct M) (simp_all add: fold_mset_insert fun_left_comm)

lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
proof (induct M)
  case empty then show ?case by simp
next
  case (add M x)
  have "M + {#x#} + N = (M + N) + {#x#}"
    by (simp add: ac_simps)
  with add show ?case by (simp add: fold_mset_insert fold_mset_fun_left_comm)
qed

lemma fold_mset_fusion:
  assumes "comp_fun_commute g"
    and *: "\<And>x y. h (g x y) = f x (h y)"
  shows "h (fold_mset g w A) = fold_mset f (h w) A"
proof -
  interpret comp_fun_commute g by (fact assms)
  from * show ?thesis by (induct A) auto
qed

end

text \<open>
  A note on code generation: When defining some function containing a
  subterm @{term "fold_mset F"}, code generation is not automatic. When
  interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
  would be code thms for @{const fold_mset} become thms like
  @{term "fold_mset F z {#} = z"} where \<open>F\<close> is not a pattern but
  contains defined symbols, i.e.\ is not a code thm. Hence a separate
  constant with its own code thms needs to be introduced for \<open>F\<close>. See the image operator below.
\<close>


subsection \<open>Image\<close>

definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where
  "image_mset f = fold_mset (plus \<circ> single \<circ> f) {#}"

lemma comp_fun_commute_mset_image: "comp_fun_commute (plus \<circ> single \<circ> f)"
proof
qed (simp add: ac_simps fun_eq_iff)

lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
  by (simp add: image_mset_def)

lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
proof -
  interpret comp_fun_commute "plus \<circ> single \<circ> f"
    by (fact comp_fun_commute_mset_image)
  show ?thesis by (simp add: image_mset_def)
qed

lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
proof -
  interpret comp_fun_commute "plus \<circ> single \<circ> f"
    by (fact comp_fun_commute_mset_image)
  show ?thesis by (induct N) (simp_all add: image_mset_def ac_simps)
qed

corollary image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
  by simp

lemma set_image_mset [simp]: "set_mset (image_mset f M) = image f (set_mset M)"
  by (induct M) simp_all

lemma size_image_mset [simp]: "size (image_mset f M) = size M"
  by (induct M) simp_all

lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
  by (cases M) auto

syntax (ASCII)
  "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ :# _#})")
syntax
  "_comprehension_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"  ("({#_/. _ \<in># _#})")
translations
  "{#e. x \<in># M#}" \<rightleftharpoons> "CONST image_mset (\<lambda>x. e) M"

syntax (ASCII)
  "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ :# _./ _#})")
syntax
  "_comprehension_mset'" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"  ("({#_/ | _ \<in># _./ _#})")
translations
  "{#e | x\<in>#M. P#}" \<rightharpoonup> "{#e. x \<in># {# x\<in>#M. P#}#}"

text \<open>
  This allows to write not just filters like @{term "{#x\<in>#M. x<c#}"}
  but also images like @{term "{#x+x. x\<in>#M #}"} and @{term [source]
  "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
  @{term "{#x+x|x\<in>#M. x<c#}"}.
\<close>

lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
by (metis set_image_mset)

functor image_mset: image_mset
proof -
  fix f g show "image_mset f \<circ> image_mset g = image_mset (f \<circ> g)"
  proof
    fix A
    show "(image_mset f \<circ> image_mset g) A = image_mset (f \<circ> g) A"
      by (induct A) simp_all
  qed
  show "image_mset id = id"
  proof
    fix A
    show "image_mset id A = id A"
      by (induct A) simp_all
  qed
qed

declare
  image_mset.id [simp]
  image_mset.identity [simp]

lemma image_mset_id[simp]: "image_mset id x = x"
  unfolding id_def by auto

lemma image_mset_cong: "(\<And>x. x \<in># M \<Longrightarrow> f x = g x) \<Longrightarrow> {#f x. x \<in># M#} = {#g x. x \<in># M#}"
  by (induct M) auto

lemma image_mset_cong_pair:
  "(\<forall>x y. (x, y) \<in># M \<longrightarrow> f x y = g x y) \<Longrightarrow> {#f x y. (x, y) \<in># M#} = {#g x y. (x, y) \<in># M#}"
  by (metis image_mset_cong split_cong)


subsection \<open>Further conversions\<close>

primrec mset :: "'a list \<Rightarrow> 'a multiset" where
  "mset [] = {#}" |
  "mset (a # x) = mset x + {# a #}"

lemma in_multiset_in_set:
  "x \<in># mset xs \<longleftrightarrow> x \<in> set xs"
  by (induct xs) simp_all

lemma count_mset:
  "count (mset xs) x = length (filter (\<lambda>y. x = y) xs)"
  by (induct xs) simp_all

lemma mset_zero_iff[simp]: "(mset x = {#}) = (x = [])"
  by (induct x) auto

lemma mset_zero_iff_right[simp]: "({#} = mset x) = (x = [])"
by (induct x) auto

lemma set_mset_mset[simp]: "set_mset (mset x) = set x"
by (induct x) auto

lemma set_mset_comp_mset [simp]: "set_mset \<circ> mset = set"
  by (simp add: fun_eq_iff)

lemma size_mset [simp]: "size (mset xs) = length xs"
  by (induct xs) simp_all

lemma mset_append [simp]: "mset (xs @ ys) = mset xs + mset ys"
  by (induct xs arbitrary: ys) (auto simp: ac_simps)

lemma mset_filter: "mset (filter P xs) = {#x \<in># mset xs. P x #}"
  by (induct xs) simp_all

lemma mset_rev [simp]:
  "mset (rev xs) = mset xs"
  by (induct xs) simp_all

lemma surj_mset: "surj mset"
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 distinct_count_atmost_1:
  "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
proof (induct x)
  case Nil then show ?case by simp
next
  case (Cons x xs) show ?case (is "?lhs \<longleftrightarrow> ?rhs")
  proof
    assume ?lhs then show ?rhs using Cons by simp
  next
    assume ?rhs then have "x \<notin> set xs"
      by (simp split: if_splits)
    moreover from \<open>?rhs\<close> have "(\<forall>a. count (mset xs) a =
       (if a \<in> set xs then 1 else 0))"
      by (auto split: if_splits simp add: count_eq_zero_iff)
    ultimately show ?lhs using Cons by simp
  qed
qed

lemma mset_eq_setD:
  assumes "mset xs = mset ys"
  shows "set xs = set ys"
proof -
  from assms have "set_mset (mset xs) = set_mset (mset ys)"
    by simp
  then show ?thesis by simp
qed

lemma set_eq_iff_mset_eq_distinct:
  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
    (set x = set y) = (mset x = mset y)"
by (auto simp: multiset_eq_iff distinct_count_atmost_1)

lemma set_eq_iff_mset_remdups_eq:
   "(set x = set y) = (mset (remdups x) = mset (remdups y))"
apply (rule iffI)
apply (simp add: set_eq_iff_mset_eq_distinct[THEN iffD1])
apply (drule distinct_remdups [THEN distinct_remdups
      [THEN set_eq_iff_mset_eq_distinct [THEN iffD2]]])
apply simp
done

lemma mset_compl_union [simp]: "mset [x\<leftarrow>xs. P x] + mset [x\<leftarrow>xs. \<not>P x] = mset xs"
  by (induct xs) (auto simp: ac_simps)

lemma nth_mem_mset: "i < length ls \<Longrightarrow> (ls ! i) \<in># mset ls"
proof (induct ls arbitrary: i)
  case Nil
  then show ?case by simp
next
  case Cons
  then show ?case by (cases i) auto
qed

lemma mset_remove1[simp]: "mset (remove1 a xs) = mset xs - {#a#}"
  by (induct xs) (auto simp add: multiset_eq_iff)

lemma mset_eq_length:
  assumes "mset xs = mset ys"
  shows "length xs = length ys"
  using assms by (metis size_mset)

lemma mset_eq_length_filter:
  assumes "mset xs = mset ys"
  shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"
  using assms by (metis count_mset)

lemma fold_multiset_equiv:
  assumes f: "\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
    and equiv: "mset xs = mset ys"
  shows "List.fold f xs = List.fold f ys"
  using f equiv [symmetric]
proof (induct xs arbitrary: ys)
  case Nil
  then show ?case by simp
next
  case (Cons x xs)
  then have *: "set ys = set (x # xs)"
    by (blast dest: mset_eq_setD)
  have "\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
    by (rule Cons.prems(1)) (simp_all add: *)
  moreover from * have "x \<in> set ys"
    by simp
  ultimately have "List.fold f ys = List.fold f (remove1 x ys) \<circ> f x"
    by (fact fold_remove1_split)
  moreover from Cons.prems have "List.fold f xs = List.fold f (remove1 x ys)"
    by (auto intro: Cons.hyps)
  ultimately show ?case by simp
qed

lemma mset_insort [simp]: "mset (insort x xs) = mset xs + {#x#}"
  by (induct xs) (simp_all add: ac_simps)

lemma mset_map: "mset (map f xs) = image_mset f (mset xs)"
  by (induct xs) simp_all

global_interpretation mset_set: folding "\<lambda>x M. {#x#} + M" "{#}"
  defines mset_set = "folding.F (\<lambda>x M. {#x#} + M) {#}"
  by standard (simp add: fun_eq_iff ac_simps)

lemma count_mset_set [simp]:
  "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> count (mset_set A) x = 1" (is "PROP ?P")
  "\<not> finite A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?Q")
  "x \<notin> A \<Longrightarrow> count (mset_set A) x = 0" (is "PROP ?R")
proof -
  have *: "count (mset_set A) x = 0" if "x \<notin> A" for A
  proof (cases "finite A")
    case False then show ?thesis by simp
  next
    case True from True \<open>x \<notin> A\<close> show ?thesis by (induct A) auto
  qed
  then show "PROP ?P" "PROP ?Q" "PROP ?R"
  by (auto elim!: Set.set_insert)
qed \<comment> \<open>TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}\<close>

lemma elem_mset_set[simp, intro]: "finite A \<Longrightarrow> x \<in># mset_set A \<longleftrightarrow> x \<in> A"
  by (induct A rule: finite_induct) simp_all

context linorder
begin

definition sorted_list_of_multiset :: "'a multiset \<Rightarrow> 'a list"
where
  "sorted_list_of_multiset M = fold_mset insort [] M"

lemma sorted_list_of_multiset_empty [simp]:
  "sorted_list_of_multiset {#} = []"
  by (simp add: sorted_list_of_multiset_def)

lemma sorted_list_of_multiset_singleton [simp]:
  "sorted_list_of_multiset {#x#} = [x]"
proof -
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  show ?thesis by (simp add: sorted_list_of_multiset_def)
qed

lemma sorted_list_of_multiset_insert [simp]:
  "sorted_list_of_multiset (M + {#x#}) = List.insort x (sorted_list_of_multiset M)"
proof -
  interpret comp_fun_commute insort by (fact comp_fun_commute_insort)
  show ?thesis by (simp add: sorted_list_of_multiset_def)
qed

end

lemma mset_sorted_list_of_multiset [simp]:
  "mset (sorted_list_of_multiset M) = M"
by (induct M) simp_all

lemma sorted_list_of_multiset_mset [simp]:
  "sorted_list_of_multiset (mset xs) = sort xs"
by (induct xs) simp_all

lemma finite_set_mset_mset_set[simp]:
  "finite A \<Longrightarrow> set_mset (mset_set A) = A"
by (induct A rule: finite_induct) simp_all

lemma infinite_set_mset_mset_set:
  "\<not> finite A \<Longrightarrow> set_mset (mset_set A) = {}"
by simp

lemma set_sorted_list_of_multiset [simp]:
  "set (sorted_list_of_multiset M) = set_mset M"
by (induct M) (simp_all add: set_insort)

lemma sorted_list_of_mset_set [simp]:
  "sorted_list_of_multiset (mset_set A) = sorted_list_of_set A"
by (cases "finite A") (induct A rule: finite_induct, simp_all add: ac_simps)


subsection \<open>Replicate operation\<close>

definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
  "replicate_mset n x = ((op + {#x#}) ^^ n) {#}"

lemma replicate_mset_0[simp]: "replicate_mset 0 x = {#}"
  unfolding replicate_mset_def by simp

lemma replicate_mset_Suc[simp]: "replicate_mset (Suc n) x = replicate_mset n x + {#x#}"
  unfolding replicate_mset_def by (induct n) (auto intro: add.commute)

lemma in_replicate_mset[simp]: "x \<in># replicate_mset n y \<longleftrightarrow> n > 0 \<and> x = y"
  unfolding replicate_mset_def by (induct n) auto

lemma count_replicate_mset[simp]: "count (replicate_mset n x) y = (if y = x then n else 0)"
  unfolding replicate_mset_def by (induct n) simp_all

lemma set_mset_replicate_mset_subset[simp]: "set_mset (replicate_mset n x) = (if n = 0 then {} else {x})"
  by (auto split: if_splits)

lemma size_replicate_mset[simp]: "size (replicate_mset n M) = n"
  by (induct n, simp_all)

lemma count_le_replicate_mset_le: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<subseteq># M"
  by (auto simp add: assms mset_less_eqI) (metis count_replicate_mset subseteq_mset_def)

lemma filter_eq_replicate_mset: "{#y \<in># D. y = x#} = replicate_mset (count D x) x"
  by (induct D) simp_all

lemma replicate_count_mset_eq_filter_eq:
  "replicate (count (mset xs) k) k = filter (HOL.eq k) xs"
  by (induct xs) auto

lemma replicate_mset_eq_empty_iff [simp]:
  "replicate_mset n a = {#} \<longleftrightarrow> n = 0"
  by (induct n) simp_all

lemma replicate_mset_eq_iff:
  "replicate_mset m a = replicate_mset n b \<longleftrightarrow>
    m = 0 \<and> n = 0 \<or> m = n \<and> a = b"
  by (auto simp add: multiset_eq_iff)


subsection \<open>Big operators\<close>

no_notation times (infixl "*" 70)
no_notation Groups.one ("1")

locale comm_monoid_mset = comm_monoid
begin

definition F :: "'a multiset \<Rightarrow> 'a"
  where eq_fold: "F M = fold_mset f 1 M"

lemma empty [simp]: "F {#} = 1"
  by (simp add: eq_fold)

lemma singleton [simp]: "F {#x#} = x"
proof -
  interpret comp_fun_commute
    by standard (simp add: fun_eq_iff left_commute)
  show ?thesis by (simp add: eq_fold)
qed

lemma union [simp]: "F (M + N) = F M * F N"
proof -
  interpret comp_fun_commute f
    by standard (simp add: fun_eq_iff left_commute)
  show ?thesis
    by (induct N) (simp_all add: left_commute eq_fold)
qed

end

lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
  by standard (simp add: add_ac comp_def)

declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]

lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (op +) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
  by (induct NN) auto

notation times (infixl "*" 70)
notation Groups.one ("1")

context comm_monoid_add
begin

sublocale msetsum: comm_monoid_mset plus 0
  defines msetsum = msetsum.F ..

lemma (in semiring_1) msetsum_replicate_mset [simp]:
  "msetsum (replicate_mset n a) = of_nat n * a"
  by (induct n) (simp_all add: algebra_simps)

lemma setsum_unfold_msetsum:
  "setsum f A = msetsum (image_mset f (mset_set A))"
  by (cases "finite A") (induct A rule: finite_induct, simp_all)

end

lemma msetsum_diff:
  fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
  shows "N \<subseteq># M \<Longrightarrow> msetsum (M - N) = msetsum M - msetsum N"
  by (metis add_diff_cancel_right' msetsum.union subset_mset.diff_add)

lemma size_eq_msetsum: "size M = msetsum (image_mset (\<lambda>_. 1) M)"
proof (induct M)
  case empty then show ?case by simp
next
  case (add M x) then show ?case
    by (cases "x \<in> set_mset M")
      (simp_all add: size_multiset_overloaded_eq setsum.distrib setsum.delta' insert_absorb not_in_iff)
qed

syntax (ASCII)
  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
syntax
  "_msetsum_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3\<Sum>_\<in>#_. _)" [0, 51, 10] 10)
translations
  "\<Sum>i \<in># A. b" \<rightleftharpoons> "CONST msetsum (CONST image_mset (\<lambda>i. b) A)"

abbreviation Union_mset :: "'a multiset multiset \<Rightarrow> 'a multiset"  ("\<Union>#_" [900] 900)
  where "\<Union># MM \<equiv> msetsum MM" -- \<open>FIXME ambiguous notation --
    could likewise refer to @{text "\<Squnion>#"}\<close>

lemma set_mset_Union_mset[simp]: "set_mset (\<Union># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
  by (induct MM) auto

lemma in_Union_mset_iff[iff]: "x \<in># \<Union># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
  by (induct MM) auto

lemma count_setsum:
  "count (setsum f A) x = setsum (\<lambda>a. count (f a) x) A"
  by (induct A rule: infinite_finite_induct) simp_all

lemma setsum_eq_empty_iff:
  assumes "finite A"
  shows "setsum f A = {#} \<longleftrightarrow> (\<forall>a\<in>A. f a = {#})"
  using assms by induct simp_all

context comm_monoid_mult
begin

sublocale msetprod: comm_monoid_mset times 1
  defines msetprod = msetprod.F ..

lemma msetprod_empty:
  "msetprod {#} = 1"
  by (fact msetprod.empty)

lemma msetprod_singleton:
  "msetprod {#x#} = x"
  by (fact msetprod.singleton)

lemma msetprod_Un:
  "msetprod (A + B) = msetprod A * msetprod B"
  by (fact msetprod.union)

lemma msetprod_replicate_mset [simp]:
  "msetprod (replicate_mset n a) = a ^ n"
  by (induct n) (simp_all add: ac_simps)

lemma setprod_unfold_msetprod:
  "setprod f A = msetprod (image_mset f (mset_set A))"
  by (cases "finite A") (induct A rule: finite_induct, simp_all)

lemma msetprod_multiplicity:
  "msetprod M = setprod (\<lambda>x. x ^ count M x) (set_mset M)"
  by (simp add: fold_mset_def setprod.eq_fold msetprod.eq_fold funpow_times_power comp_def)

end

syntax (ASCII)
  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
syntax
  "_msetprod_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3\<Prod>_\<in>#_. _)" [0, 51, 10] 10)
translations
  "\<Prod>i \<in># A. b" \<rightleftharpoons> "CONST msetprod (CONST image_mset (\<lambda>i. b) A)"

lemma (in comm_semiring_1) dvd_msetprod:
  assumes "x \<in># A"
  shows "x dvd msetprod A"
proof -
  from assms have "A = (A - {#x#}) + {#x#}" by simp
  then obtain B where "A = B + {#x#}" ..
  then show ?thesis by simp
qed

lemma (in semidom) msetprod_zero_iff [iff]:
  "msetprod A = 0 \<longleftrightarrow> 0 \<in># A"
  by (induct A) auto

lemma (in semidom_divide) msetprod_diff:
  assumes "B \<subseteq># A" and "0 \<notin># B"
  shows "msetprod (A - B) = msetprod A div msetprod B"
proof -
  from assms obtain C where "A = B + C"
    by (metis subset_mset.add_diff_inverse)
  with assms show ?thesis by simp
qed

lemma (in semidom_divide) msetprod_minus:
  assumes "a \<in># A" and "a \<noteq> 0"
  shows "msetprod (A - {#a#}) = msetprod A div a"
  using assms msetprod_diff [of "{#a#}" A]
    by (auto simp add: single_subset_iff)

lemma (in normalization_semidom) normalized_msetprodI:
  assumes "\<And>a. a \<in># A \<Longrightarrow> normalize a = a"
  shows "normalize (msetprod A) = msetprod A"
  using assms by (induct A) (simp_all add: normalize_mult)


subsection \<open>Alternative representations\<close>

subsubsection \<open>Lists\<close>

context linorder
begin

lemma mset_insort [simp]:
  "mset (insort_key k x xs) = {#x#} + mset xs"
  by (induct xs) (simp_all add: ac_simps)

lemma mset_sort [simp]:
  "mset (sort_key k xs) = mset xs"
  by (induct xs) (simp_all add: ac_simps)

text \<open>
  This lemma shows which properties suffice to show that a function
  \<open>f\<close> with \<open>f xs = ys\<close> behaves like sort.
\<close>

lemma properties_for_sort_key:
  assumes "mset ys = mset xs"
    and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"
    and "sorted (map f ys)"
  shows "sort_key f xs = ys"
  using assms
proof (induct xs arbitrary: ys)
  case Nil then show ?case by simp
next
  case (Cons x xs)
  from Cons.prems(2) have
    "\<forall>k \<in> set ys. filter (\<lambda>x. f k = f x) (remove1 x ys) = filter (\<lambda>x. f k = f x) xs"
    by (simp add: filter_remove1)
  with Cons.prems have "sort_key f xs = remove1 x ys"
    by (auto intro!: Cons.hyps simp add: sorted_map_remove1)
  moreover from Cons.prems have "x \<in># mset ys"
    by auto
  then have "x \<in> set ys"
    by simp
  ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
qed

lemma properties_for_sort:
  assumes multiset: "mset ys = mset xs"
    and "sorted ys"
  shows "sort xs = ys"
proof (rule properties_for_sort_key)
  from multiset show "mset ys = mset xs" .
  from \<open>sorted ys\<close> show "sorted (map (\<lambda>x. x) ys)" by simp
  from multiset have "length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)" for k
    by (rule mset_eq_length_filter)
  then have "replicate (length (filter (\<lambda>y. k = y) ys)) k =
    replicate (length (filter (\<lambda>x. k = x) xs)) k" for k
    by simp
  then show "k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs" for k
    by (simp add: replicate_length_filter)
qed

lemma sort_key_inj_key_eq:
  assumes mset_equal: "mset xs = mset ys"
    and "inj_on f (set xs)"
    and "sorted (map f ys)"
  shows "sort_key f xs = ys"
proof (rule properties_for_sort_key)
  from mset_equal
  show "mset ys = mset xs" by simp
  from \<open>sorted (map f ys)\<close>
  show "sorted (map f ys)" .
  show "[x\<leftarrow>ys . f k = f x] = [x\<leftarrow>xs . f k = f x]" if "k \<in> set ys" for k
  proof -
    from mset_equal
    have set_equal: "set xs = set ys" by (rule mset_eq_setD)
    with that have "insert k (set ys) = set ys" by auto
    with \<open>inj_on f (set xs)\<close> have inj: "inj_on f (insert k (set ys))"
      by (simp add: set_equal)
    from inj have "[x\<leftarrow>ys . f k = f x] = filter (HOL.eq k) ys"
      by (auto intro!: inj_on_filter_key_eq)
    also have "\<dots> = replicate (count (mset ys) k) k"
      by (simp add: replicate_count_mset_eq_filter_eq)
    also have "\<dots> = replicate (count (mset xs) k) k"
      using mset_equal by simp
    also have "\<dots> = filter (HOL.eq k) xs"
      by (simp add: replicate_count_mset_eq_filter_eq)
    also have "\<dots> = [x\<leftarrow>xs . f k = f x]"
      using inj by (auto intro!: inj_on_filter_key_eq [symmetric] simp add: set_equal)
    finally show ?thesis .
  qed
qed

lemma sort_key_eq_sort_key:
  assumes "mset xs = mset ys"
    and "inj_on f (set xs)"
  shows "sort_key f xs = sort_key f ys"
  by (rule sort_key_inj_key_eq) (simp_all add: assms)

lemma sort_key_by_quicksort:
  "sort_key f xs = sort_key f [x\<leftarrow>xs. f x < f (xs ! (length xs div 2))]
    @ [x\<leftarrow>xs. f x = f (xs ! (length xs div 2))]
    @ sort_key f [x\<leftarrow>xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
proof (rule properties_for_sort_key)
  show "mset ?rhs = mset ?lhs"
    by (rule multiset_eqI) (auto simp add: mset_filter)
  show "sorted (map f ?rhs)"
    by (auto simp add: sorted_append intro: sorted_map_same)
next
  fix l
  assume "l \<in> set ?rhs"
  let ?pivot = "f (xs ! (length xs div 2))"
  have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto
  have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"
    unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
  with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp
  have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto
  then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =
    [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp
  note *** = this [of "op <"] this [of "op >"] this [of "op ="]
  show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"
  proof (cases "f l" ?pivot rule: linorder_cases)
    case less
    then have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto
    with less show ?thesis
      by (simp add: filter_sort [symmetric] ** ***)
  next
    case equal then show ?thesis
      by (simp add: * less_le)
  next
    case greater
    then have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto
    with greater show ?thesis
      by (simp add: filter_sort [symmetric] ** ***)
  qed
qed

lemma sort_by_quicksort:
  "sort xs = sort [x\<leftarrow>xs. x < xs ! (length xs div 2)]
    @ [x\<leftarrow>xs. x = xs ! (length xs div 2)]
    @ sort [x\<leftarrow>xs. x > xs ! (length xs div 2)]" (is "sort ?lhs = ?rhs")
  using sort_key_by_quicksort [of "\<lambda>x. x", symmetric] by simp

text \<open>A stable parametrized quicksort\<close>

definition part :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b list \<Rightarrow> 'b list \<times> 'b list \<times> 'b list" where
  "part f pivot xs = ([x \<leftarrow> xs. f x < pivot], [x \<leftarrow> xs. f x = pivot], [x \<leftarrow> xs. pivot < f x])"

lemma part_code [code]:
  "part f pivot [] = ([], [], [])"
  "part f pivot (x # xs) = (let (lts, eqs, gts) = part f pivot xs; x' = f x in
     if x' < pivot then (x # lts, eqs, gts)
     else if x' > pivot then (lts, eqs, x # gts)
     else (lts, x # eqs, gts))"
  by (auto simp add: part_def Let_def split_def)

lemma sort_key_by_quicksort_code [code]:
  "sort_key f xs =
    (case xs of
      [] \<Rightarrow> []
    | [x] \<Rightarrow> xs
    | [x, y] \<Rightarrow> (if f x \<le> f y then xs else [y, x])
    | _ \<Rightarrow>
        let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
        in sort_key f lts @ eqs @ sort_key f gts)"
proof (cases xs)
  case Nil then show ?thesis by simp
next
  case (Cons _ ys) note hyps = Cons show ?thesis
  proof (cases ys)
    case Nil with hyps show ?thesis by simp
  next
    case (Cons _ zs) note hyps = hyps Cons show ?thesis
    proof (cases zs)
      case Nil with hyps show ?thesis by auto
    next
      case Cons
      from sort_key_by_quicksort [of f xs]
      have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
        in sort_key f lts @ eqs @ sort_key f gts)"
      by (simp only: split_def Let_def part_def fst_conv snd_conv)
      with hyps Cons show ?thesis by (simp only: list.cases)
    qed
  qed
qed

end

hide_const (open) part

lemma mset_remdups_le: "mset (remdups xs) \<subseteq># mset xs"
  by (induct xs) (auto intro: subset_mset.order_trans)

lemma mset_update:
  "i < length ls \<Longrightarrow> mset (ls[i := v]) = mset 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_mset)
      done
  qed
qed

lemma mset_swap:
  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
    mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
  by (cases "i = j") (simp_all add: mset_update nth_mem_mset)


subsection \<open>The multiset order\<close>

subsubsection \<open>Well-foundedness\<close>

definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
  "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
      (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r)}"

definition mult :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
  "mult r = (mult1 r)\<^sup>+"

lemma mult1I:
  assumes "M = M0 + {#a#}" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
  shows "(N, M) \<in> mult1 r"
  using assms unfolding mult1_def by blast

lemma mult1E:
  assumes "(N, M) \<in> mult1 r"
  obtains a M0 K where "M = M0 + {#a#}" "N = M0 + K" "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
  using assms unfolding mult1_def by blast

lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
by (simp add: mult1_def)

lemma less_add:
  assumes mult1: "(N, M0 + {#a#}) \<in> mult1 r"
  shows
    "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
     (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
proof -
  let ?r = "\<lambda>K a. \<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r"
  let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
  obtain a' M0' K where M0: "M0 + {#a#} = M0' + {#a'#}"
    and N: "N = M0' + K"
    and r: "?r K a'"
    using mult1 unfolding mult1_def by auto
  show ?thesis (is "?case1 \<or> ?case2")
  proof -
    from M0 consider "M0 = M0'" "a = a'"
      | K' where "M0 = K' + {#a'#}" "M0' = K' + {#a#}"
      by atomize_elim (simp only: add_eq_conv_ex)
    then show ?thesis
    proof cases
      case 1
      with N r have "?r K a \<and> N = M0 + K" by simp
      then have ?case2 ..
      then show ?thesis ..
    next
      case 2
      from N 2(2) have n: "N = K' + K + {#a#}" by (simp add: ac_simps)
      with r 2(1) have "?R (K' + K) M0" by blast
      with n have ?case1 by (simp add: mult1_def)
      then show ?thesis ..
    qed
  qed
qed

lemma all_accessible:
  assumes "wf r"
  shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
proof
  let ?R = "mult1 r"
  let ?W = "Wellfounded.acc ?R"
  {
    fix M M0 a
    assume M0: "M0 \<in> ?W"
      and wf_hyp: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
      and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W"
    have "M0 + {#a#} \<in> ?W"
    proof (rule accI [of "M0 + {#a#}"])
      fix N
      assume "(N, M0 + {#a#}) \<in> ?R"
      then consider M where "(M, M0) \<in> ?R" "N = M + {#a#}"
        | K where "\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r" "N = M0 + K"
        by atomize_elim (rule less_add)
      then show "N \<in> ?W"
      proof cases
        case 1
        from acc_hyp have "(M, M0) \<in> ?R \<longrightarrow> M + {#a#} \<in> ?W" ..
        from this and \<open>(M, M0) \<in> ?R\<close> have "M + {#a#} \<in> ?W" ..
        then show "N \<in> ?W" by (simp only: \<open>N = M + {#a#}\<close>)
      next
        case 2
        from this(1) have "M0 + K \<in> ?W"
        proof (induct K)
          case empty
          from M0 show "M0 + {#} \<in> ?W" by simp
        next
          case (add K x)
          from add.prems have "(x, a) \<in> r" by simp
          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: add.assoc)
        qed
        then show "N \<in> ?W" by (simp only: 2(2))
      qed
    qed
  } note tedious_reasoning = this

  show "M \<in> ?W" for M
  proof (induct M)
    show "{#} \<in> ?W"
    proof (rule accI)
      fix b assume "(b, {#}) \<in> ?R"
      with not_less_empty show "b \<in> ?W" by contradiction
    qed

    fix M a assume "M \<in> ?W"
    from \<open>wf r\<close> have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
    proof induct
      fix a
      assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
      show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
      proof
        fix M assume "M \<in> ?W"
        then show "M + {#a#} \<in> ?W"
          by (rule acc_induct) (rule tedious_reasoning [OF _ r])
      qed
    qed
    from this and \<open>M \<in> ?W\<close> show "M + {#a#} \<in> ?W" ..
  qed
qed

theorem wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
by (rule acc_wfI) (rule all_accessible)

theorem wf_mult: "wf r \<Longrightarrow> wf (mult r)"
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)


subsubsection \<open>Closure-free presentation\<close>

text \<open>One direction.\<close>

lemma mult_implies_one_step:
  "trans r \<Longrightarrow> (M, N) \<in> mult r \<Longrightarrow>
    \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
    (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)"
apply (unfold mult_def mult1_def)
apply (erule converse_trancl_induct, clarify)
 apply (rule_tac x = M0 in exI, simp, clarify)
apply (case_tac "a \<in># K")
 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: add.assoc [symmetric])
 apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong)
 apply (simp add: diff_union_single_conv)
 apply (simp (no_asm_use) add: trans_def)
 apply (metis (no_types, hide_lams) Multiset.diff_right_commute Un_iff diff_single_trivial multi_drop_mem_not_eq)
apply (subgoal_tac "a \<in># I")
 apply (rule_tac x = "I - {#a#}" in exI)
 apply (rule_tac x = "J + {#a#}" in exI)
 apply (rule_tac x = "K + Ka" in exI)
 apply (rule conjI)
  apply (simp add: multiset_eq_iff split: nat_diff_split)
 apply (rule conjI)
  apply (drule_tac f = "\<lambda>M. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
  apply (simp add: multiset_eq_iff split: nat_diff_split)
 apply (simp (no_asm_use) add: trans_def)
apply (subgoal_tac "a \<in># (M0 + {#a#})")
 apply (simp_all add: not_in_iff)
 apply blast
 apply (metis add.comm_neutral add_diff_cancel_right' count_eq_zero_iff diff_single_trivial multi_self_add_other_not_self plus_multiset.rep_eq)
done

lemma one_step_implies_mult_aux:
  "\<forall>I J K. size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r)
    \<longrightarrow> (I + K, I + J) \<in> mult r"
apply (induct n)
 apply auto
apply (frule size_eq_Suc_imp_eq_union, clarify)
apply (rename_tac "J'", simp)
apply (erule notE, auto)
apply (case_tac "J' = {#}")
 apply (simp add: mult_def)
 apply (rule r_into_trancl)
 apply (simp add: mult1_def, blast)
txt \<open>Now we know @{term "J' \<noteq> {#}"}.\<close>
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
apply (erule_tac P = "\<forall>k \<in> set_mset K. P k" for P in rev_mp)
apply (erule ssubst)
apply (simp add: Ball_def, auto)
apply (subgoal_tac
  "((I + {# x \<in># K. (x, a) \<in> r #}) + {# x \<in># K. (x, a) \<notin> r #},
    (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r")
 prefer 2
 apply force
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)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
apply (simp add: ac_simps)
done

lemma one_step_implies_mult:
  "trans r \<Longrightarrow> J \<noteq> {#} \<Longrightarrow> \<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r
    \<Longrightarrow> (I + K, I + J) \<in> mult r"
using one_step_implies_mult_aux by blast


subsubsection \<open>Partial-order properties\<close>

lemma (in order) mult1_lessE:
  assumes "(N, M) \<in> mult1 {(a, b). a < b}"
  obtains a M0 K where "M = M0 + {#a#}" "N = M0 + K"
    "a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> b < a"
proof -
  from assms obtain a M0 K where "M = M0 + {#a#}" "N = M0 + K"
    "\<And>b. b \<in># K \<Longrightarrow> b < a" by (blast elim: mult1E)
  moreover from this(3) [of a] have "a \<notin># K" by auto
  ultimately show thesis by (auto intro: that)
qed

definition less_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subset>#" 50)
  where "M' #\<subset># M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"

definition le_multiset :: "'a::order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "#\<subseteq>#" 50)
  where "M' #\<subseteq># M \<longleftrightarrow> M' #\<subset># M \<or> M' = M"

notation (ASCII)
  less_multiset (infix "#<#" 50) and
  le_multiset (infix "#<=#" 50)

interpretation multiset_order: order le_multiset less_multiset
proof -
  have irrefl: "\<not> M #\<subset># M" for M :: "'a multiset"
  proof
    assume "M #\<subset># M"
    then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)
    have "trans {(x'::'a, x). x' < x}"
      by (rule transI) simp
    moreover note MM
    ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
      \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})"
      by (rule mult_implies_one_step)
    then obtain I J K where "M = I + J" and "M = I + K"
      and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast
    then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto
    have "finite (set_mset K)" by simp
    moreover note **
    ultimately have "set_mset K = {}"
      by (induct rule: finite_induct) (auto intro: order_less_trans)
    with * show False by simp
  qed
  have trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># N" for K M N :: "'a multiset"
    unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
  show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset"
    by standard (auto simp add: le_multiset_def irrefl dest: trans)
qed -- \<open>FIXME avoid junk stemming from type class interpretation\<close>

lemma mult_less_irrefl [elim!]:
  fixes M :: "'a::order multiset"
  shows "M #\<subset># M \<Longrightarrow> R"
  by simp


subsubsection \<open>Monotonicity of multiset union\<close>

lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
apply (unfold mult1_def)
apply auto
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
apply (simp add: add.assoc)
done

lemma union_less_mono2: "B #\<subset># D \<Longrightarrow> C + B #\<subset># C + (D::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
apply (erule trancl_induct)
 apply (blast intro: mult1_union)
apply (blast intro: mult1_union trancl_trans)
done

lemma union_less_mono1: "B #\<subset># D \<Longrightarrow> B + C #\<subset># D + (C::'a::order multiset)"
apply (subst add.commute [of B C])
apply (subst add.commute [of D C])
apply (erule union_less_mono2)
done

lemma union_less_mono:
  fixes A B C D :: "'a::order multiset"
  shows "A #\<subset># C \<Longrightarrow> B #\<subset># D \<Longrightarrow> A + B #\<subset># C + D"
  by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)

interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
  by standard (auto simp add: le_multiset_def intro: union_less_mono2)


subsubsection \<open>Termination proofs with multiset orders\<close>

lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
  and multi_member_this: "x \<in># {# x #} + XS"
  and multi_member_last: "x \<in># {# x #}"
  by auto

definition "ms_strict = mult pair_less"
definition "ms_weak = ms_strict \<union> Id"

lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
by (auto intro: wf_mult1 wf_trancl simp: mult_def)

lemma smsI:
  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
  unfolding ms_strict_def
by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)

lemma wmsI:
  "(set_mset A, set_mset B) \<in> max_strict \<or> A = {#} \<and> B = {#}
  \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
unfolding ms_weak_def ms_strict_def
by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)

inductive pw_leq
where
  pw_leq_empty: "pw_leq {#} {#}"
| pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"

lemma pw_leq_lstep:
  "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
by (drule pw_leq_step) (rule pw_leq_empty, simp)

lemma pw_leq_split:
  assumes "pw_leq X Y"
  shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
  using assms
proof induct
  case pw_leq_empty thus ?case by auto
next
  case (pw_leq_step x y X Y)
  then obtain A B Z where
    [simp]: "X = A + Z" "Y = B + Z"
      and 1[simp]: "(set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#})"
    by auto
  from pw_leq_step consider "x = y" | "(x, y) \<in> pair_less"
    unfolding pair_leq_def by auto
  thus ?case
  proof cases
    case [simp]: 1
    have "{#x#} + X = A + ({#y#}+Z) \<and> {#y#} + Y = B + ({#y#}+Z) \<and>
      ((set_mset A, set_mset B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
      by (auto simp: ac_simps)
    thus ?thesis by blast
  next
    case 2
    let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
    have "{#x#} + X = ?A' + Z"
      "{#y#} + Y = ?B' + Z"
      by (auto simp add: ac_simps)
    moreover have
      "(set_mset ?A', set_mset ?B') \<in> max_strict"
      using 1 2 unfolding max_strict_def
      by (auto elim!: max_ext.cases)
    ultimately show ?thesis by blast
  qed
qed

lemma
  assumes pwleq: "pw_leq Z Z'"
  shows ms_strictI: "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
    and ms_weakI1:  "(set_mset A, set_mset B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
    and ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
proof -
  from pw_leq_split[OF pwleq]
  obtain A' B' Z''
    where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
    and mx_or_empty: "(set_mset A', set_mset B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
    by blast
  {
    assume max: "(set_mset A, set_mset B) \<in> max_strict"
    from mx_or_empty
    have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
    proof
      assume max': "(set_mset A', set_mset B') \<in> max_strict"
      with max have "(set_mset (A + A'), set_mset (B + B')) \<in> max_strict"
        by (auto simp: max_strict_def intro: max_ext_additive)
      thus ?thesis by (rule smsI)
    next
      assume [simp]: "A' = {#} \<and> B' = {#}"
      show ?thesis by (rule smsI) (auto intro: max)
    qed
    thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add: ac_simps)
    thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
  }
  from mx_or_empty
  have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
  thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:ac_simps)
qed

lemma empty_neutral: "{#} + x = x" "x + {#} = x"
and nonempty_plus: "{# x #} + rs \<noteq> {#}"
and nonempty_single: "{# x #} \<noteq> {#}"
by auto

setup \<open>
  let
    fun msetT T = Type (@{type_name multiset}, [T]);

    fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)
      | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
      | mk_mset T (x :: xs) =
            Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
                  mk_mset T [x] $ mk_mset T xs

    fun mset_member_tac ctxt m i =
      if m <= 0 then
        resolve_tac ctxt @{thms multi_member_this} i ORELSE
        resolve_tac ctxt @{thms multi_member_last} i
      else
        resolve_tac ctxt @{thms multi_member_skip} i THEN mset_member_tac ctxt (m - 1) i

    fun mset_nonempty_tac ctxt =
      resolve_tac ctxt @{thms nonempty_plus} ORELSE'
      resolve_tac ctxt @{thms nonempty_single}

    fun regroup_munion_conv ctxt =
      Function_Lib.regroup_conv ctxt @{const_abbrev Mempty} @{const_name plus}
        (map (fn t => t RS eq_reflection) (@{thms ac_simps} @ @{thms empty_neutral}))

    fun unfold_pwleq_tac ctxt i =
      (resolve_tac ctxt @{thms pw_leq_step} i THEN (fn st => unfold_pwleq_tac ctxt (i + 1) st))
        ORELSE (resolve_tac ctxt @{thms pw_leq_lstep} i)
        ORELSE (resolve_tac ctxt @{thms pw_leq_empty} i)

    val set_mset_simps = [@{thm set_mset_empty}, @{thm set_mset_single}, @{thm set_mset_union},
                        @{thm Un_insert_left}, @{thm Un_empty_left}]
  in
    ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset
    {
      msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
      mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
      mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_mset_simps,
      smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},
      reduction_pair = @{thm ms_reduction_pair}
    })
  end
\<close>


subsection \<open>Legacy theorem bindings\<close>

lemmas multi_count_eq = multiset_eq_iff [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_left_imp_eq)

lemma mset_less_trans: "(M::'a multiset) \<subset># K \<Longrightarrow> K \<subset># N \<Longrightarrow> M \<subset># N"
  by (fact subset_mset.less_trans)

lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
  by (fact subset_mset.inf.commute)

lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
  by (fact subset_mset.inf.assoc [symmetric])

lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
  by (fact subset_mset.inf.left_commute)

lemmas multiset_inter_ac =
  multiset_inter_commute
  multiset_inter_assoc
  multiset_inter_left_commute

lemma mult_less_not_refl: "\<not> M #\<subset># (M::'a::order multiset)"
  by (fact multiset_order.less_irrefl)

lemma mult_less_trans: "K #\<subset># M \<Longrightarrow> M #\<subset># N \<Longrightarrow> K #\<subset># (N::'a::order multiset)"
  by (fact multiset_order.less_trans)

lemma mult_less_not_sym: "M #\<subset># N \<Longrightarrow> \<not> N #\<subset># (M::'a::order multiset)"
  by (fact multiset_order.less_not_sym)

lemma mult_less_asym: "M #\<subset># N \<Longrightarrow> (\<not> P \<Longrightarrow> N #\<subset># (M::'a::order multiset)) \<Longrightarrow> P"
  by (fact multiset_order.less_asym)

declaration \<open>
  let
    fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T])) (Const _ $ t') =
          let
            val (maybe_opt, ps) =
              Nitpick_Model.dest_plain_fun t'
              ||> op ~~
              ||> map (apsnd (snd o HOLogic.dest_number))
            fun elems_for t =
              (case AList.lookup (op =) ps t of
                SOME n => replicate n t
              | NONE => [Const (maybe_name, elem_T --> elem_T) $ t])
          in
            (case maps elems_for (all_values elem_T) @
                 (if maybe_opt then [Const (Nitpick_Model.unrep_mixfix (), elem_T)] else []) of
              [] => Const (@{const_name zero_class.zero}, T)
            | ts =>
                foldl1 (fn (t1, t2) =>
                    Const (@{const_name plus_class.plus}, T --> T --> T) $ t1 $ t2)
                  (map (curry (op $) (Const (@{const_name single}, elem_T --> T))) ts))
          end
      | multiset_postproc _ _ _ _ t = t
  in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end
\<close>


subsection \<open>Naive implementation using lists\<close>

code_datatype mset

lemma [code]: "{#} = mset []"
  by simp

lemma [code]: "{#x#} = mset [x]"
  by simp

lemma union_code [code]: "mset xs + mset ys = mset (xs @ ys)"
  by simp

lemma [code]: "image_mset f (mset xs) = mset (map f xs)"
  by (simp add: mset_map)

lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
  by (simp add: mset_filter)

lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"
  by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute)

lemma [code]:
  "mset xs #\<inter> mset ys =
    mset (snd (fold (\<lambda>x (ys, zs).
      if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
proof -
  have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
    if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
      (mset xs #\<inter> mset ys) + mset zs"
    by (induct xs arbitrary: ys)
      (auto simp add: inter_add_right1 inter_add_right2 ac_simps)
  then show ?thesis by simp
qed

lemma [code]:
  "mset xs #\<union> mset ys =
    mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
proof -
  have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
      (mset xs #\<union> mset ys) + mset zs"
    by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
  then show ?thesis by simp
qed

declare in_multiset_in_set [code_unfold]

lemma [code]: "count (mset xs) x = fold (\<lambda>y. if x = y then Suc else id) xs 0"
proof -
  have "\<And>n. fold (\<lambda>y. if x = y then Suc else id) xs n = count (mset xs) x + n"
    by (induct xs) simp_all
  then show ?thesis by simp
qed

declare set_mset_mset [code]

declare sorted_list_of_multiset_mset [code]

lemma [code]: \<comment> \<open>not very efficient, but representation-ignorant!\<close>
  "mset_set A = mset (sorted_list_of_set A)"
  apply (cases "finite A")
  apply simp_all
  apply (induct A rule: finite_induct)
  apply (simp_all add: add.commute)
  done

declare size_mset [code]

fun ms_lesseq_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
  "ms_lesseq_impl [] ys = Some (ys \<noteq> [])"
| "ms_lesseq_impl (Cons x xs) ys = (case List.extract (op = x) ys of
     None \<Rightarrow> None
   | Some (ys1,_,ys2) \<Rightarrow> ms_lesseq_impl xs (ys1 @ ys2))"

lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<subseteq># mset ys) \<and>
  (ms_lesseq_impl xs ys = Some True \<longleftrightarrow> mset xs \<subset># mset ys) \<and>
  (ms_lesseq_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
proof (induct xs arbitrary: ys)
  case (Nil ys)
  show ?case by (auto simp: mset_less_empty_nonempty)
next
  case (Cons x xs ys)
  show ?case
  proof (cases "List.extract (op = x) ys")
    case None
    hence x: "x \<notin> set ys" by (simp add: extract_None_iff)
    {
      assume "mset (x # xs) \<subseteq># mset ys"
      from set_mset_mono[OF this] x have False by simp
    } note nle = this
    moreover
    {
      assume "mset (x # xs) \<subset># mset ys"
      hence "mset (x # xs) \<subseteq># mset ys" by auto
      from nle[OF this] have False .
    }
    ultimately show ?thesis using None by auto
  next
    case (Some res)
    obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
    note Some = Some[unfolded res]
    from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
    hence id: "mset ys = mset (ys1 @ ys2) + {#x#}"
      by (auto simp: ac_simps)
    show ?thesis unfolding ms_lesseq_impl.simps
      unfolding Some option.simps split
      unfolding id
      using Cons[of "ys1 @ ys2"]
      unfolding subset_mset_def subseteq_mset_def by auto
  qed
qed

lemma [code]: "mset xs \<subseteq># mset ys \<longleftrightarrow> ms_lesseq_impl xs ys \<noteq> None"
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)

lemma [code]: "mset xs \<subset># mset ys \<longleftrightarrow> ms_lesseq_impl xs ys = Some True"
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)

instantiation multiset :: (equal) equal
begin

definition
  [code del]: "HOL.equal A (B :: 'a multiset) \<longleftrightarrow> A = B"
lemma [code]: "HOL.equal (mset xs) (mset ys) \<longleftrightarrow> ms_lesseq_impl xs ys = Some False"
  unfolding equal_multiset_def
  using ms_lesseq_impl[of xs ys] by (cases "ms_lesseq_impl xs ys", auto)

instance
  by standard (simp add: equal_multiset_def)

end

lemma [code]: "msetsum (mset xs) = listsum xs"
  by (induct xs) (simp_all add: add.commute)

lemma [code]: "msetprod (mset xs) = fold times xs 1"
proof -
  have "\<And>x. fold times xs x = msetprod (mset xs) * x"
    by (induct xs) (simp_all add: mult.assoc)
  then show ?thesis by simp
qed

text \<open>
  Exercise for the casual reader: add implementations for @{const le_multiset}
  and @{const less_multiset} (multiset order).
\<close>

text \<open>Quickcheck generators\<close>

definition (in term_syntax)
  msetify :: "'a::typerep list \<times> (unit \<Rightarrow> Code_Evaluation.term)
    \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
  [code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {\<cdot>} xs"

notation fcomp (infixl "\<circ>>" 60)
notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation multiset :: (random) random
begin

definition
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"

instance ..

end

no_notation fcomp (infixl "\<circ>>" 60)
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation multiset :: (full_exhaustive) full_exhaustive
begin

definition full_exhaustive_multiset :: "('a multiset \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
where
  "full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (\<lambda>xs. f (msetify xs)) i"

instance ..

end

hide_const (open) msetify


subsection \<open>BNF setup\<close>

definition rel_mset where
  "rel_mset R X Y \<longleftrightarrow> (\<exists>xs ys. mset xs = X \<and> mset ys = Y \<and> list_all2 R xs ys)"

lemma mset_zip_take_Cons_drop_twice:
  assumes "length xs = length ys" "j \<le> length xs"
  shows "mset (zip (take j xs @ x # drop j xs) (take j ys @ y # drop j ys)) =
    mset (zip xs ys) + {#(x, y)#}"
  using assms
proof (induct xs ys arbitrary: x y j rule: list_induct2)
  case Nil
  thus ?case
    by simp
next
  case (Cons x xs y ys)
  thus ?case
  proof (cases "j = 0")
    case True
    thus ?thesis
      by simp
  next
    case False
    then obtain k where k: "j = Suc k"
      by (cases j) simp
    hence "k \<le> length xs"
      using Cons.prems by auto
    hence "mset (zip (take k xs @ x # drop k xs) (take k ys @ y # drop k ys)) =
      mset (zip xs ys) + {#(x, y)#}"
      by (rule Cons.hyps(2))
    thus ?thesis
      unfolding k by (auto simp: add.commute union_lcomm)
  qed
qed

lemma ex_mset_zip_left:
  assumes "length xs = length ys" "mset xs' = mset xs"
  shows "\<exists>ys'. length ys' = length xs' \<and> mset (zip xs' ys') = mset (zip xs ys)"
using assms
proof (induct xs ys arbitrary: xs' rule: list_induct2)
  case Nil
  thus ?case
    by auto
next
  case (Cons x xs y ys xs')
  obtain j where j_len: "j < length xs'" and nth_j: "xs' ! j = x"
    by (metis Cons.prems in_set_conv_nth list.set_intros(1) mset_eq_setD)

  def xsa \<equiv> "take j xs' @ drop (Suc j) xs'"
  have "mset xs' = {#x#} + mset xsa"
    unfolding xsa_def using j_len nth_j
    by (metis (no_types) ab_semigroup_add_class.add_ac(1) append_take_drop_id Cons_nth_drop_Suc
      mset.simps(2) union_code add.commute)
  hence ms_x: "mset xsa = mset xs"
    by (metis Cons.prems add.commute add_right_imp_eq mset.simps(2))
  then obtain ysa where
    len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
    using Cons.hyps(2) by blast

  def ys' \<equiv> "take j ysa @ y # drop j ysa"
  have xs': "xs' = take j xsa @ x # drop j xsa"
    using ms_x j_len nth_j Cons.prems xsa_def
    by (metis append_eq_append_conv append_take_drop_id diff_Suc_Suc Cons_nth_drop_Suc length_Cons
      length_drop size_mset)
  have j_len': "j \<le> length xsa"
    using j_len xs' xsa_def
    by (metis add_Suc_right append_take_drop_id length_Cons length_append less_eq_Suc_le not_less)
  have "length ys' = length xs'"
    unfolding ys'_def using Cons.prems len_a ms_x
    by (metis add_Suc_right append_take_drop_id length_Cons length_append mset_eq_length)
  moreover have "mset (zip xs' ys') = mset (zip (x # xs) (y # ys))"
    unfolding xs' ys'_def
    by (rule trans[OF mset_zip_take_Cons_drop_twice])
      (auto simp: len_a ms_a j_len' add.commute)
  ultimately show ?case
    by blast
qed

lemma list_all2_reorder_left_invariance:
  assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
  shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
proof -
  have len: "length xs = length ys"
    using rel list_all2_conv_all_nth by auto
  obtain ys' where
    len': "length xs' = length ys'" and ms_xy: "mset (zip xs' ys') = mset (zip xs ys)"
    using len ms_x by (metis ex_mset_zip_left)
  have "list_all2 R xs' ys'"
    using assms(1) len' ms_xy unfolding list_all2_iff by (blast dest: mset_eq_setD)
  moreover have "mset ys' = mset ys"
    using len len' ms_xy map_snd_zip mset_map by metis
  ultimately show ?thesis
    by blast
qed

lemma ex_mset: "\<exists>xs. mset xs = X"
  by (induct X) (simp, metis mset.simps(2))

inductive pred_mset :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> bool"
where
  "pred_mset P {#}"
| "\<lbrakk>P a; pred_mset P M\<rbrakk> \<Longrightarrow> pred_mset P (M + {#a#})"

bnf "'a multiset"
  map: image_mset
  sets: set_mset
  bd: natLeq
  wits: "{#}"
  rel: rel_mset
  pred: pred_mset
proof -
  show "image_mset id = id"
    by (rule image_mset.id)
  show "image_mset (g \<circ> f) = image_mset g \<circ> image_mset f" for f g
    unfolding comp_def by (rule ext) (simp add: comp_def image_mset.compositionality)
  show "(\<And>z. z \<in> set_mset X \<Longrightarrow> f z = g z) \<Longrightarrow> image_mset f X = image_mset g X" for f g X
    by (induct X) simp_all
  show "set_mset \<circ> image_mset f = op ` f \<circ> set_mset" for f
    by auto
  show "card_order natLeq"
    by (rule natLeq_card_order)
  show "BNF_Cardinal_Arithmetic.cinfinite natLeq"
    by (rule natLeq_cinfinite)
  show "ordLeq3 (card_of (set_mset X)) natLeq" for X
    by transfer
      (auto intro!: ordLess_imp_ordLeq simp: finite_iff_ordLess_natLeq[symmetric] multiset_def)
  show "rel_mset R OO rel_mset S \<le> rel_mset (R OO S)" for R S
    unfolding rel_mset_def[abs_def] OO_def
    apply clarify
    subgoal for X Z Y xs ys' ys zs
      apply (drule list_all2_reorder_left_invariance [where xs = ys' and ys = zs and xs' = ys])
      apply (auto intro: list_all2_trans)
      done
    done
  show "rel_mset R =
    (\<lambda>x y. \<exists>z. set_mset z \<subseteq> {(x, y). R x y} \<and>
    image_mset fst z = x \<and> image_mset snd z = y)" for R
    unfolding rel_mset_def[abs_def]
    apply (rule ext)+
    apply safe
     apply (rule_tac x = "mset (zip xs ys)" in exI;
       auto simp: in_set_zip list_all2_iff mset_map[symmetric])
    apply (rename_tac XY)
    apply (cut_tac X = XY in ex_mset)
    apply (erule exE)
    apply (rename_tac xys)
    apply (rule_tac x = "map fst xys" in exI)
    apply (auto simp: mset_map)
    apply (rule_tac x = "map snd xys" in exI)
    apply (auto simp: mset_map list_all2I subset_eq zip_map_fst_snd)
    done
  show "z \<in> set_mset {#} \<Longrightarrow> False" for z
    by auto
  show "pred_mset P = (\<lambda>x. Ball (set_mset x) P)" for P
  proof (intro ext iffI)
    fix x
    assume "pred_mset P x"
    then show "Ball (set_mset x) P" by (induct pred: pred_mset; simp)
  next
    fix x
    assume "Ball (set_mset x) P"
    then show "pred_mset P x" by (induct x; auto intro: pred_mset.intros)
  qed
qed

inductive rel_mset'
where
  Zero[intro]: "rel_mset' R {#} {#}"
| Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (M + {#a#}) (N + {#b#})"

lemma rel_mset_Zero: "rel_mset R {#} {#}"
unfolding rel_mset_def Grp_def by auto

declare multiset.count[simp]
declare Abs_multiset_inverse[simp]
declare multiset.count_inverse[simp]
declare union_preserves_multiset[simp]

lemma rel_mset_Plus:
  assumes ab: "R a b"
    and MN: "rel_mset R M N"
  shows "rel_mset R (M + {#a#}) (N + {#b#})"
proof -
  have "\<exists>ya. image_mset fst y + {#a#} = image_mset fst ya \<and>
    image_mset snd y + {#b#} = image_mset snd ya \<and>
    set_mset ya \<subseteq> {(x, y). R x y}"
    if "R a b" and "set_mset y \<subseteq> {(x, y). R x y}" for y
    using that by (intro exI[of _ "y + {#(a,b)#}"]) auto
  thus ?thesis
  using assms
  unfolding multiset.rel_compp_Grp Grp_def by blast
qed

lemma rel_mset'_imp_rel_mset: "rel_mset' R M N \<Longrightarrow> rel_mset R M N"
  by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)

lemma rel_mset_size: "rel_mset R M N \<Longrightarrow> size M = size N"
  unfolding multiset.rel_compp_Grp Grp_def by auto

lemma multiset_induct2[case_names empty addL addR]:
  assumes empty: "P {#} {#}"
    and addL: "\<And>M N a. P M N \<Longrightarrow> P (M + {#a#}) N"
    and addR: "\<And>M N a. P M N \<Longrightarrow> P M (N + {#a#})"
  shows "P M N"
apply(induct N rule: multiset_induct)
  apply(induct M rule: multiset_induct, rule empty, erule addL)
  apply(induct M rule: multiset_induct, erule addR, erule addR)
done

lemma multiset_induct2_size[consumes 1, case_names empty add]:
  assumes c: "size M = size N"
    and empty: "P {#} {#}"
    and add: "\<And>M N a b. P M N \<Longrightarrow> P (M + {#a#}) (N + {#b#})"
  shows "P M N"
  using c
proof (induct M arbitrary: N rule: measure_induct_rule[of size])
  case (less M)
  show ?case
  proof(cases "M = {#}")
    case True hence "N = {#}" using less.prems by auto
    thus ?thesis using True empty by auto
  next
    case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
    have "N \<noteq> {#}" using False less.prems by auto
    then obtain N1 b where N: "N = N1 + {#b#}" by (metis multi_nonempty_split)
    have "size M1 = size N1" using less.prems unfolding M N by auto
    thus ?thesis using M N less.hyps add by auto
  qed
qed

lemma msed_map_invL:
  assumes "image_mset f (M + {#a#}) = N"
  shows "\<exists>N1. N = N1 + {#f a#} \<and> image_mset f M = N1"
proof -
  have "f a \<in># N"
    using assms multiset.set_map[of f "M + {#a#}"] by auto
  then obtain N1 where N: "N = N1 + {#f a#}" using multi_member_split by metis
  have "image_mset f M = N1" using assms unfolding N by simp
  thus ?thesis using N by blast
qed

lemma msed_map_invR:
  assumes "image_mset f M = N + {#b#}"
  shows "\<exists>M1 a. M = M1 + {#a#} \<and> f a = b \<and> image_mset f M1 = N"
proof -
  obtain a where a: "a \<in># M" and fa: "f a = b"
    using multiset.set_map[of f M] unfolding assms
    by (metis image_iff union_single_eq_member)
  then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
  have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
  thus ?thesis using M fa by blast
qed

lemma msed_rel_invL:
  assumes "rel_mset R (M + {#a#}) N"
  shows "\<exists>N1 b. N = N1 + {#b#} \<and> R a b \<and> rel_mset R M N1"
proof -
  obtain K where KM: "image_mset fst K = M + {#a#}"
    and KN: "image_mset snd K = N" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
    using assms
    unfolding multiset.rel_compp_Grp Grp_def by auto
  obtain K1 ab where K: "K = K1 + {#ab#}" and a: "fst ab = a"
    and K1M: "image_mset fst K1 = M" using msed_map_invR[OF KM] by auto
  obtain N1 where N: "N = N1 + {#snd ab#}" and K1N1: "image_mset snd K1 = N1"
    using msed_map_invL[OF KN[unfolded K]] by auto
  have Rab: "R a (snd ab)" using sK a unfolding K by auto
  have "rel_mset R M N1" using sK K1M K1N1
    unfolding K multiset.rel_compp_Grp Grp_def by auto
  thus ?thesis using N Rab by auto
qed

lemma msed_rel_invR:
  assumes "rel_mset R M (N + {#b#})"
  shows "\<exists>M1 a. M = M1 + {#a#} \<and> R a b \<and> rel_mset R M1 N"
proof -
  obtain K where KN: "image_mset snd K = N + {#b#}"
    and KM: "image_mset fst K = M" and sK: "set_mset K \<subseteq> {(a, b). R a b}"
    using assms
    unfolding multiset.rel_compp_Grp Grp_def by auto
  obtain K1 ab where K: "K = K1 + {#ab#}" and b: "snd ab = b"
    and K1N: "image_mset snd K1 = N" using msed_map_invR[OF KN] by auto
  obtain M1 where M: "M = M1 + {#fst ab#}" and K1M1: "image_mset fst K1 = M1"
    using msed_map_invL[OF KM[unfolded K]] by auto
  have Rab: "R (fst ab) b" using sK b unfolding K by auto
  have "rel_mset R M1 N" using sK K1N K1M1
    unfolding K multiset.rel_compp_Grp Grp_def by auto
  thus ?thesis using M Rab by auto
qed

lemma rel_mset_imp_rel_mset':
  assumes "rel_mset R M N"
  shows "rel_mset' R M N"
using assms proof(induct M arbitrary: N rule: measure_induct_rule[of size])
  case (less M)
  have c: "size M = size N" using rel_mset_size[OF less.prems] .
  show ?case
  proof(cases "M = {#}")
    case True hence "N = {#}" using c by simp
    thus ?thesis using True rel_mset'.Zero by auto
  next
    case False then obtain M1 a where M: "M = M1 + {#a#}" by (metis multi_nonempty_split)
    obtain N1 b where N: "N = N1 + {#b#}" and R: "R a b" and ms: "rel_mset R M1 N1"
      using msed_rel_invL[OF less.prems[unfolded M]] by auto
    have "rel_mset' R M1 N1" using less.hyps[of M1 N1] ms unfolding M by simp
    thus ?thesis using rel_mset'.Plus[of R a b, OF R] unfolding M N by simp
  qed
qed

lemma rel_mset_rel_mset': "rel_mset R M N = rel_mset' R M N"
  using rel_mset_imp_rel_mset' rel_mset'_imp_rel_mset by auto

text \<open>The main end product for @{const rel_mset}: inductive characterization:\<close>
lemmas rel_mset_induct[case_names empty add, induct pred: rel_mset] =
  rel_mset'.induct[unfolded rel_mset_rel_mset'[symmetric]]


subsection \<open>Size setup\<close>

lemma multiset_size_o_map: "size_multiset g \<circ> image_mset f = size_multiset (g \<circ> f)"
  apply (rule ext)
  subgoal for x by (induct x) auto
  done

setup \<open>
  BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}
    @{thm size_multiset_overloaded_def}
    @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
      size_union}
    @{thms multiset_size_o_map}
\<close>

hide_const (open) wcount

end