diff --git a/src/HOL/Library/Multiset.thy b/src/HOL/Library/Multiset.thy
--- a/src/HOL/Library/Multiset.thy
+++ b/src/HOL/Library/Multiset.thy
@@ -1,4092 +1,4111 @@
 (*  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
     Author:     Martin Desharnais, MPI-INF Saarbruecken
 *)
 
 section \<open>(Finite) Multisets\<close>
 
 theory Multiset
   imports Cancellation
 begin
 
 subsection \<open>The type of multisets\<close>
 
 typedef 'a multiset = \<open>{f :: 'a \<Rightarrow> nat. finite {x. f x > 0}}\<close>
   morphisms count Abs_multiset
 proof
   show \<open>(\<lambda>x. 0::nat) \<in> {f. finite {x. f x > 0}}\<close>
     by simp
 qed
 
 setup_lifting type_definition_multiset
 
 lemma count_Abs_multiset:
   \<open>count (Abs_multiset f) = f\<close> if \<open>finite {x. f x > 0}\<close>
   by (rule Abs_multiset_inverse) (simp add: that)
 
 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>\<open>multiset\<close>.\<close>
 
 lemma diff_preserves_multiset:
   \<open>finite {x. 0 < M x - N x}\<close> if \<open>finite {x. 0 < M x}\<close> for M N :: \<open>'a \<Rightarrow> nat\<close>
   using that by (rule rev_finite_subset) auto
 
 lemma filter_preserves_multiset:
   \<open>finite {x. 0 < (if P x then M x else 0)}\<close> if \<open>finite {x. 0 < M x}\<close> for M N :: \<open>'a \<Rightarrow> nat\<close>
   using that by (rule rev_finite_subset) auto
 
 lemmas in_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 :: \<open>'a multiset\<close>
   is \<open>\<lambda>a. 0\<close>
   by simp
 
 abbreviation empty_mset :: \<open>'a multiset\<close> (\<open>{#}\<close>)
   where \<open>empty_mset \<equiv> 0\<close>
 
 lift_definition plus_multiset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>M N a. M a + N a\<close>
   by simp
 
 lift_definition minus_multiset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>M N a. M a - N a\<close>
   by (rule diff_preserves_multiset)
 
 instance
   by (standard; transfer) (simp_all add: fun_eq_iff)
 
 end
 
 context
 begin
 
 qualified definition is_empty :: "'a multiset \<Rightarrow> bool" where
   [code_abbrev]: "is_empty A \<longleftrightarrow> A = {#}"
 
 end
 
 lemma add_mset_in_multiset:
   \<open>finite {x. 0 < (if x = a then Suc (M x) else M x)}\<close>
   if \<open>finite {x. 0 < M x}\<close>
   using that by (simp add: flip: insert_Collect)
 
 lift_definition add_mset :: "'a \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" is
   "\<lambda>a M b. if b = a then Suc (M b) else M b"
 by (rule add_mset_in_multiset)
 
 syntax
   "_multiset" :: "args \<Rightarrow> 'a multiset"    ("{#(_)#}")
 translations
   "{#x, xs#}" == "CONST add_mset x {#xs#}"
   "{#x#}" == "CONST add_mset x {#}"
 
 lemma count_empty [simp]: "count {#} a = 0"
   by (simp add: zero_multiset.rep_eq)
 
 lemma count_add_mset [simp]:
   "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
   by (simp add: add_mset.rep_eq)
 
 lemma count_single: "count {#b#} a = (if b = a then 1 else 0)"
   by simp
 
 lemma
   add_mset_not_empty [simp]: \<open>add_mset a A \<noteq> {#}\<close> and
   empty_not_add_mset [simp]: "{#} \<noteq> add_mset a A"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_add_mset_same_iff [simp]:
   "add_mset a A = add_mset a B \<longleftrightarrow> A = B"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_commute:
   "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
   by (auto simp: multiset_eq_iff)
 
 
 subsection \<open>Basic operations\<close>
 
 subsubsection \<open>Conversion to set and membership\<close>
 
 definition set_mset :: \<open>'a multiset \<Rightarrow> 'a set\<close>
   where \<open>set_mset M = {x. count M x > 0}\<close>
 
 abbreviation member_mset :: \<open>'a \<Rightarrow> 'a multiset \<Rightarrow> bool\<close>
   where \<open>member_mset a M \<equiv> a \<in> set_mset M\<close>
 
 notation
   member_mset  (\<open>'(\<in>#')\<close>) and
   member_mset  (\<open>(_/ \<in># _)\<close> [50, 51] 50)
 
 notation  (ASCII)
   member_mset  (\<open>'(:#')\<close>) and
   member_mset  (\<open>(_/ :# _)\<close> [50, 51] 50)
 
 abbreviation not_member_mset :: \<open>'a \<Rightarrow> 'a multiset \<Rightarrow> bool\<close>
   where \<open>not_member_mset a M \<equiv> a \<notin> set_mset M\<close>
 
 notation
   not_member_mset  (\<open>'(\<notin>#')\<close>) and
   not_member_mset  (\<open>(_/ \<notin># _)\<close> [50, 51] 50)
 
 notation  (ASCII)
   not_member_mset  (\<open>'(~:#')\<close>) and
   not_member_mset  (\<open>(_/ ~:# _)\<close> [50, 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)"
 
 print_translation \<open>
  [Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Multiset.Ball\<close> \<^syntax_const>\<open>_MBall\<close>,
   Syntax_Trans.preserve_binder_abs2_tr' \<^const_syntax>\<open>Multiset.Bex\<close> \<^syntax_const>\<open>_MBex\<close>]
 \<close> \<comment> \<open>to avoid eta-contraction of body\<close>
 
 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:
   "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
 
 lemma set_mset_add_mset_insert [simp]: \<open>set_mset (add_mset a A) = insert a (set_mset A)\<close>
   by (auto simp flip: count_greater_eq_Suc_zero_iff split: if_splits)
 
 lemma multiset_nonemptyE [elim]:
   assumes "A \<noteq> {#}"
   obtains x where "x \<in># A"
 proof -
   have "\<exists>x. x \<in># A" by (rule ccontr) (insert assms, auto)
   with that show ?thesis by blast
 qed
 
 
 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
 
 lemma union_mset_add_mset_left [simp]:
   "add_mset a A + B = add_mset a (A + B)"
   by (auto simp: multiset_eq_iff)
 
 lemma union_mset_add_mset_right [simp]:
   "A + add_mset a B = add_mset a (A + B)"
   by (auto simp: multiset_eq_iff)
 
 lemma add_mset_add_single: \<open>add_mset a A = A + {#a#}\<close>
   by (subst union_mset_add_mset_right, subst add.comm_neutral) standard
 
 
 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 add_mset_diff_bothsides:
   \<open>add_mset a M - add_mset a A = M - A\<close>
   by (auto simp: multiset_eq_iff)
 
 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: "A - A = {#}"
   by (fact Groups.diff_cancel)
 
 lemma diff_union_cancelR: "M + N - N = (M::'a multiset)"
   by (fact add_diff_cancel_right')
 
 lemma diff_union_cancelL: "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 [simp]: "x \<in># M \<Longrightarrow> add_mset x (M - {#x#}) = M"
   by (clarsimp simp: multiset_eq_iff)
 
 lemma insert_DiffM2: "x \<in># M \<Longrightarrow> (M - {#x#}) + {#x#} = M"
   by simp
 
 lemma diff_union_swap: "a \<noteq> b \<Longrightarrow> add_mset b (M - {#a#}) = add_mset b M - {#a#}"
   by (auto simp add: multiset_eq_iff)
 
 lemma diff_add_mset_swap [simp]: "b \<notin># A \<Longrightarrow> add_mset b M - A = add_mset b (M - A)"
   by (auto simp add: multiset_eq_iff simp: not_in_iff)
 
 lemma diff_union_swap2 [simp]: "y \<in># M \<Longrightarrow> add_mset x M - {#y#} = add_mset x (M - {#y#})"
   by (metis add_mset_diff_bothsides diff_union_swap diff_zero insert_DiffM)
 
 lemma diff_diff_add_mset [simp]: "(M::'a multiset) - N - P = M - (N + P)"
   by (rule diff_diff_add)
 
 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 = add_mset a B"
 proof -
   from assms have "A = add_mset 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>Min and Max\<close>
 
 abbreviation Min_mset :: "'a::linorder multiset \<Rightarrow> 'a" where
 "Min_mset m \<equiv> Min (set_mset m)"
 
 abbreviation Max_mset :: "'a::linorder multiset \<Rightarrow> 'a" where
 "Max_mset m \<equiv> Max (set_mset m)"
 
 
 subsubsection \<open>Equality of multisets\<close>
 
 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 = add_mset x M \<longleftrightarrow> False"
   by (auto simp add: multiset_eq_iff)
 
 lemma add_mset_remove_trivial [simp]: \<open>add_mset x M - {#x#} = M\<close>
   by (auto simp: 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 = add_mset x N"
   by auto
 
 lemma union_single_eq_diff: "add_mset x M = N \<Longrightarrow> M = N - {#x#}"
   unfolding add_mset_add_single[of _ M] by (fact add_implies_diff)
 
 lemma union_single_eq_member: "add_mset x M = N \<Longrightarrow> x \<in># N"
   by auto
 
 lemma add_mset_remove_trivial_If:
   "add_mset a (N - {#a#}) = (if a \<in># N then N else add_mset a N)"
   by (simp add: diff_single_trivial)
 
 lemma add_mset_remove_trivial_eq: \<open>N = add_mset a (N - {#a#}) \<longleftrightarrow> a \<in># N\<close>
   by (auto simp: add_mset_remove_trivial_If)
 
 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:
   "add_mset a M = add_mset b N \<longleftrightarrow> M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#})"
   (is "?lhs \<longleftrightarrow> ?rhs")
 (* shorter: by (simp add: multiset_eq_iff) fastforce *)
 proof
   show ?lhs if ?rhs
     using that
     by (auto simp add: add_mset_commute[of a b])
   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># add_mset b N" by (rule union_single_eq_member)
     with False have "a \<in># N" by auto
     moreover from \<open>?lhs\<close> have "M = add_mset b N - {#a#}" by (rule union_single_eq_diff)
     moreover note False
     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"])
   qed
 qed
 
 lemma add_mset_eq_single [iff]: "add_mset b M = {#a#} \<longleftrightarrow> b = a \<and> M = {#}"
   by (auto simp: add_eq_conv_diff)
 
 lemma single_eq_add_mset [iff]: "{#a#} = add_mset b M \<longleftrightarrow> b = a \<and> M = {#}"
   by (auto simp: add_eq_conv_diff)
 
 lemma insert_noteq_member:
   assumes BC: "add_mset b B = add_mset c C"
    and bnotc: "b \<noteq> c"
   shows "c \<in># B"
 proof -
   have "c \<in># add_mset c C" by simp
   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
   then have "c \<in># add_mset b B" using BC by simp
   then show "c \<in># B" using nc by simp
 qed
 
 lemma add_eq_conv_ex:
   "(add_mset a M = add_mset b N) =
     (M = N \<and> a = b \<or> (\<exists>K. M = add_mset b K \<and> N = add_mset a K))"
   by (auto simp add: add_eq_conv_diff)
 
 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = add_mset x A"
   by (rule exI [where x = "M - {#x#}"]) simp
 
 lemma multiset_add_sub_el_shuffle:
   assumes "c \<in># B"
     and "b \<noteq> c"
   shows "add_mset b (B - {#c#}) = add_mset b B - {#c#}"
 proof -
   from \<open>c \<in># B\<close> obtain A where B: "B = add_mset c A"
     by (blast dest: multi_member_split)
   have "add_mset b A = add_mset c (add_mset b A) - {#c#}" by simp
   then have "add_mset b A = add_mset b (add_mset c A) - {#c#}"
     by (simp add: \<open>b \<noteq> c\<close>)
   then show ?thesis using B by simp
 qed
 
 lemma add_mset_eq_singleton_iff[iff]:
   "add_mset x M = {#y#} \<longleftrightarrow> M = {#} \<and> x = y"
   by auto
 
 
 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 \<longleftrightarrow> (\<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 \<longleftrightarrow> 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)
 
 global_interpretation subset_mset: ordering \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order.trans order.antisym)
 
 interpretation subset_mset: ordered_ab_semigroup_add_imp_le \<open>(+)\<close> \<open>(-)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 interpretation subset_mset: ordered_ab_semigroup_monoid_add_imp_le "(+)" 0 "(-)" "(\<subseteq>#)" "(\<subset>#)"
   by standard
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 lemma mset_subset_eqI:
   "(\<And>a. count A a \<le> count B a) \<Longrightarrow> A \<subseteq># B"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eq_count:
   "A \<subseteq># B \<Longrightarrow> count A a \<le> count B a"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eq_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 "(+)" 0 "(\<subseteq>#)" "(\<subset>#)" "(-)"
   by standard (simp, fact mset_subset_eq_exists_conv)
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 declare subset_mset.add_diff_assoc[simp] subset_mset.add_diff_assoc2[simp]
 
 lemma mset_subset_eq_mono_add_right_cancel: "(A::'a multiset) + C \<subseteq># B + C \<longleftrightarrow> A \<subseteq># B"
    by (fact subset_mset.add_le_cancel_right)
 
 lemma mset_subset_eq_mono_add_left_cancel: "C + (A::'a multiset) \<subseteq># C + B \<longleftrightarrow> A \<subseteq># B"
    by (fact subset_mset.add_le_cancel_left)
 
 lemma mset_subset_eq_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_subset_eq_add_left: "(A::'a multiset) \<subseteq># A + B"
    by simp
 
 lemma mset_subset_eq_add_right: "B \<subseteq># (A::'a multiset) + B"
    by simp
 
 lemma single_subset_iff [simp]:
   "{#a#} \<subseteq># M \<longleftrightarrow> a \<in># M"
   by (auto simp add: subseteq_mset_def Suc_le_eq)
 
 lemma mset_subset_eq_single: "a \<in># B \<Longrightarrow> {#a#} \<subseteq># B"
   by simp
 
 lemma mset_subset_eq_add_mset_cancel: \<open>add_mset a A \<subseteq># add_mset a B \<longleftrightarrow> A \<subseteq># B\<close>
   unfolding add_mset_add_single[of _ A] add_mset_add_single[of _ B]
   by (rule mset_subset_eq_mono_add_right_cancel)
 
 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_subset_eq_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_subset_eq_self[simp]:
   "(M::'a multiset) - N \<subseteq># M"
   by (simp add: subseteq_mset_def)
 
 lemma mset_subset_eqD:
   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_subsetD:
   "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
   by (auto intro: mset_subset_eqD [of A])
 
 lemma set_mset_mono:
   "A \<subseteq># B \<Longrightarrow> set_mset A \<subseteq> set_mset B"
   by (metis mset_subset_eqD subsetI)
 
 lemma mset_subset_eq_insertD:
   "add_mset x A \<subseteq># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
 apply (rule conjI)
  apply (simp add: mset_subset_eqD)
  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_subset_insertD:
   "add_mset x A \<subset># B \<Longrightarrow> x \<in># B \<and> A \<subset># B"
   by (rule mset_subset_eq_insertD) simp
 
 lemma mset_subset_of_empty[simp]: "A \<subset># {#} \<longleftrightarrow> False"
   by (simp only: subset_mset.not_less_zero)
 
 lemma empty_subset_add_mset[simp]: "{#} \<subset># add_mset x M"
   by (auto intro: subset_mset.gr_zeroI)
 
 lemma empty_le: "{#} \<subseteq># A"
   by (fact subset_mset.zero_le)
 
 lemma insert_subset_eq_iff:
   "add_mset 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:
   "add_mset a A \<subset># B \<longleftrightarrow> a \<in># B \<and> A \<subset># B - {#a#}"
   by (auto simp add: insert_subset_eq_iff subset_mset_def)
 
 lemma subset_eq_diff_conv:
   "A - C \<subseteq># B \<longleftrightarrow> A \<subseteq># B + C"
   by (simp add: subseteq_mset_def le_diff_conv)
 
 lemma multi_psub_of_add_self [simp]: "A \<subset># add_mset x A"
   by (auto simp: subset_mset_def subseteq_mset_def)
 
 lemma multi_psub_self: "A \<subset># A = False"
   by simp
 
 lemma mset_subset_add_mset [simp]: "add_mset x N \<subset># add_mset x M \<longleftrightarrow> N \<subset># M"
   unfolding add_mset_add_single[of _ N] add_mset_add_single[of _ M]
   by (fact subset_mset.add_less_cancel_right)
 
 lemma mset_subset_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
   by (auto simp: subset_mset_def elim: mset_add)
 
 lemma Diff_eq_empty_iff_mset: "A - B = {#} \<longleftrightarrow> A \<subseteq># B"
   by (auto simp: multiset_eq_iff subseteq_mset_def)
 
 lemma add_mset_subseteq_single_iff[iff]: "add_mset a M \<subseteq># {#b#} \<longleftrightarrow> M = {#} \<and> a = b"
 proof
   assume A: "add_mset a M \<subseteq># {#b#}"
   then have \<open>a = b\<close>
     by (auto dest: mset_subset_eq_insertD)
   then show "M={#} \<and> a=b"
     using A by (simp add: mset_subset_eq_add_mset_cancel)
 qed simp
 
 
 subsubsection \<open>Intersection and bounded union\<close>
 
 definition inter_mset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>  (infixl \<open>\<inter>#\<close> 70)
   where \<open>A \<inter># B = A - (A - B)\<close>
 
 lemma count_inter_mset [simp]:
   \<open>count (A \<inter># B) x = min (count A x) (count B x)\<close>
   by (simp add: inter_mset_def)
 
 (*global_interpretation subset_mset: semilattice_order \<open>(\<inter>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (simp_all add: multiset_eq_iff subseteq_mset_def subset_mset_def min_def)*)
 
 interpretation subset_mset: semilattice_inf \<open>(\<inter>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
   by standard (simp_all add: multiset_eq_iff subseteq_mset_def)
   \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 definition union_mset :: \<open>'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>  (infixl \<open>\<union>#\<close> 70)
   where \<open>A \<union># B = A + (B - A)\<close>
 
 lemma count_union_mset [simp]:
   \<open>count (A \<union># B) x = max (count A x) (count B x)\<close>
   by (simp add: union_mset_def)
 
 global_interpretation subset_mset: semilattice_neutr_order \<open>(\<union>#)\<close> \<open>{#}\<close> \<open>(\<supseteq>#)\<close> \<open>(\<supset>#)\<close>
   apply standard
       apply (simp_all add: multiset_eq_iff subseteq_mset_def subset_mset_def max_def)
    apply (auto simp add: le_antisym dest: sym)
    apply (metis nat_le_linear)+
   done
 
 interpretation subset_mset: semilattice_sup \<open>(\<union>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
 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 (\<union>#) (\<subseteq>#) (\<subset>#)"
     by standard (auto simp add: union_mset_def subseteq_mset_def)
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 interpretation subset_mset: bounded_lattice_bot "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)"
   "(\<union>#)" "{#}"
   by standard auto
     \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 
 subsubsection \<open>Additional intersection facts\<close>
 
 lemma set_mset_inter [simp]:
   "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
   by (simp only: set_mset_def) auto
 
 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[simp]: "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 inter_mset_empty_distrib_right: "A \<inter># (B + C) = {#} \<longleftrightarrow> A \<inter># B = {#} \<and> A \<inter># C = {#}"
   by (meson disjunct_not_in union_iff)
 
 lemma inter_mset_empty_distrib_left: "(A + B) \<inter># C = {#} \<longleftrightarrow> A \<inter># C = {#} \<and> B \<inter># C = {#}"
   by (meson disjunct_not_in union_iff)
 
 lemma add_mset_inter_add_mset [simp]:
   "add_mset a A \<inter># add_mset a B = add_mset a (A \<inter># B)"
   by (rule multiset_eqI) simp
 
 lemma add_mset_disjoint [simp]:
   "add_mset a A \<inter># B = {#} \<longleftrightarrow> a \<notin># B \<and> A \<inter># B = {#}"
   "{#} = add_mset a A \<inter># B \<longleftrightarrow> a \<notin># B \<and> {#} = A \<inter># B"
   by (auto simp: disjunct_not_in)
 
 lemma disjoint_add_mset [simp]:
   "B \<inter># add_mset a A = {#} \<longleftrightarrow> a \<notin># B \<and> B \<inter># A = {#}"
   "{#} = A \<inter># add_mset b B \<longleftrightarrow> b \<notin># A \<and> {#} = A \<inter># B"
   by (auto simp: disjunct_not_in)
 
 lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = M \<inter># N"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma inter_add_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = add_mset x (M \<inter># (N - {#x#}))"
   by (auto simp add: multiset_eq_iff elim: mset_add)
 
 lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = N \<inter># M"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma inter_add_right2: "x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = add_mset x ((N - {#x#}) \<inter># M)"
   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>Additional bounded union facts\<close>
 
 lemma set_mset_sup [simp]:
   \<open>set_mset (A \<union># B) = set_mset A \<union> set_mset B\<close>
   by (simp only: set_mset_def) (auto simp add: less_max_iff_disj)
 
 lemma sup_union_left1 [simp]: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># N)"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma sup_union_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># (N - {#x#}))"
   by (simp add: multiset_eq_iff)
 
 lemma sup_union_right1 [simp]: "\<not> x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x (N \<union># M)"
   by (simp add: multiset_eq_iff not_in_iff)
 
 lemma sup_union_right2: "x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x ((N - {#x#}) \<union># M)"
   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)
 
 lemma add_mset_union:
   \<open>add_mset a A \<union># add_mset a B = add_mset a (A \<union># B)\<close>
   by (auto simp: multiset_eq_iff max_def)
 
 
 subsection \<open>Replicate and repeat operations\<close>
 
 definition replicate_mset :: "nat \<Rightarrow> 'a \<Rightarrow> 'a multiset" where
   "replicate_mset n x = (add_mset 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 = add_mset x (replicate_mset n x)"
   unfolding replicate_mset_def by (induct n) (auto intro: add.commute)
 
 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) auto
 
 lift_definition repeat_mset :: \<open>nat \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset\<close>
   is \<open>\<lambda>n M a. n * M a\<close> by simp
 
 lemma count_repeat_mset [simp]: "count (repeat_mset i A) a = i * count A a"
   by transfer rule
 
 lemma repeat_mset_0 [simp]:
   \<open>repeat_mset 0 M = {#}\<close>
   by transfer simp
 
 lemma repeat_mset_Suc [simp]:
   \<open>repeat_mset (Suc n) M = M + repeat_mset n M\<close>
   by transfer simp
 
 lemma repeat_mset_right [simp]: "repeat_mset a (repeat_mset b A) = repeat_mset (a * b) A"
   by (auto simp: multiset_eq_iff left_diff_distrib')
 
 lemma left_diff_repeat_mset_distrib': \<open>repeat_mset (i - j) u = repeat_mset i u - repeat_mset j u\<close>
   by (auto simp: multiset_eq_iff left_diff_distrib')
 
 lemma left_add_mult_distrib_mset:
   "repeat_mset i u + (repeat_mset j u + k) = repeat_mset (i+j) u + k"
   by (auto simp: multiset_eq_iff add_mult_distrib)
 
 lemma repeat_mset_distrib:
   "repeat_mset (m + n) A = repeat_mset m A + repeat_mset n A"
   by (auto simp: multiset_eq_iff Nat.add_mult_distrib)
 
 lemma repeat_mset_distrib2[simp]:
   "repeat_mset n (A + B) = repeat_mset n A + repeat_mset n B"
   by (auto simp: multiset_eq_iff add_mult_distrib2)
 
 lemma repeat_mset_replicate_mset[simp]:
   "repeat_mset n {#a#} = replicate_mset n a"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_distrib_add_mset[simp]:
   "repeat_mset n (add_mset a A) = replicate_mset n a + repeat_mset n A"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_empty[simp]: "repeat_mset n {#} = {#}"
   by transfer simp
 
 
 subsubsection \<open>Simprocs\<close>
 
 lemma repeat_mset_iterate_add: \<open>repeat_mset n M = iterate_add n M\<close>
   unfolding iterate_add_def by (induction n) auto
 
 lemma mset_subseteq_add_iff1:
   "j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (repeat_mset (i-j) u + m \<subseteq># n)"
   by (auto simp add: subseteq_mset_def nat_le_add_iff1)
 
 lemma mset_subseteq_add_iff2:
   "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<subseteq># repeat_mset j u + n) = (m \<subseteq># repeat_mset (j-i) u + n)"
   by (auto simp add: subseteq_mset_def nat_le_add_iff2)
 
 lemma mset_subset_add_iff1:
   "j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<subset># repeat_mset j u + n) = (repeat_mset (i-j) u + m \<subset># n)"
   unfolding subset_mset_def repeat_mset_iterate_add
   by (simp add: iterate_add_eq_add_iff1 mset_subseteq_add_iff1[unfolded repeat_mset_iterate_add])
 
 lemma mset_subset_add_iff2:
   "i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<subset># repeat_mset j u + n) = (m \<subset># repeat_mset (j-i) u + n)"
   unfolding subset_mset_def repeat_mset_iterate_add
   by (simp add: iterate_add_eq_add_iff2 mset_subseteq_add_iff2[unfolded repeat_mset_iterate_add])
 
 ML_file \<open>multiset_simprocs.ML\<close>
 
 lemma add_mset_replicate_mset_safe[cancelation_simproc_pre]: \<open>NO_MATCH {#} M \<Longrightarrow> add_mset a M = {#a#} + M\<close>
   by simp
 
 declare repeat_mset_iterate_add[cancelation_simproc_pre]
 
 declare iterate_add_distrib[cancelation_simproc_pre]
 declare repeat_mset_iterate_add[symmetric, cancelation_simproc_post]
 
 declare add_mset_not_empty[cancelation_simproc_eq_elim]
     empty_not_add_mset[cancelation_simproc_eq_elim]
     subset_mset.le_zero_eq[cancelation_simproc_eq_elim]
     empty_not_add_mset[cancelation_simproc_eq_elim]
     add_mset_not_empty[cancelation_simproc_eq_elim]
     subset_mset.le_zero_eq[cancelation_simproc_eq_elim]
     le_zero_eq[cancelation_simproc_eq_elim]
 
 simproc_setup mseteq_cancel
   ("(l::'a multiset) + m = n" | "(l::'a multiset) = m + n" |
    "add_mset a m = n" | "m = add_mset a n" |
    "replicate_mset p a = n" | "m = replicate_mset p a" |
    "repeat_mset p m = n" | "m = repeat_mset p m") =
   \<open>fn phi => Cancel_Simprocs.eq_cancel\<close>
 
 simproc_setup msetsubset_cancel
   ("(l::'a multiset) + m \<subset># n" | "(l::'a multiset) \<subset># m + n" |
    "add_mset a m \<subset># n" | "m \<subset># add_mset a n" |
    "replicate_mset p r \<subset># n" | "m \<subset># replicate_mset p r" |
    "repeat_mset p m \<subset># n" | "m \<subset># repeat_mset p m") =
   \<open>fn phi => Multiset_Simprocs.subset_cancel_msets\<close>
 
 simproc_setup msetsubset_eq_cancel
   ("(l::'a multiset) + m \<subseteq># n" | "(l::'a multiset) \<subseteq># m + n" |
    "add_mset a m \<subseteq># n" | "m \<subseteq># add_mset a n" |
    "replicate_mset p r \<subseteq># n" | "m \<subseteq># replicate_mset p r" |
    "repeat_mset p m \<subseteq># n" | "m \<subseteq># repeat_mset p m") =
   \<open>fn phi => Multiset_Simprocs.subseteq_cancel_msets\<close>
 
 simproc_setup msetdiff_cancel
   ("((l::'a multiset) + m) - n" | "(l::'a multiset) - (m + n)" |
    "add_mset a m - n" | "m - add_mset a n" |
    "replicate_mset p r - n" | "m - replicate_mset p r" |
    "repeat_mset p m - n" | "m - repeat_mset p m") =
   \<open>fn phi => Cancel_Simprocs.diff_cancel\<close>
 
 
 subsubsection \<open>Conditionally complete lattice\<close>
 
 instantiation multiset :: (type) Inf
 begin
 
 lift_definition Inf_multiset :: "'a multiset set \<Rightarrow> 'a multiset" is
   "\<lambda>A i. if A = {} then 0 else Inf ((\<lambda>f. f i) ` A)"
 proof -
   fix A :: "('a \<Rightarrow> nat) set"
   assume *: "\<And>f. f \<in> A \<Longrightarrow> finite {x. 0 < f x}"
   show \<open>finite {i. 0 < (if A = {} then 0 else INF f\<in>A. f i)}\<close>
   proof (cases "A = {}")
     case False
     then obtain f where "f \<in> A" by blast
     hence "{i. Inf ((\<lambda>f. f i) ` A) > 0} \<subseteq> {i. f i > 0}"
       by (auto intro: less_le_trans[OF _ cInf_lower])
     moreover from \<open>f \<in> A\<close> * have "finite \<dots>" by simp
     ultimately have "finite {i. Inf ((\<lambda>f. f i) ` A) > 0}" by (rule finite_subset)
     with False show ?thesis by simp
   qed simp_all
 qed
 
 instance ..
 
 end
 
 lemma Inf_multiset_empty: "Inf {} = {#}"
   by transfer simp_all
 
 lemma count_Inf_multiset_nonempty: "A \<noteq> {} \<Longrightarrow> count (Inf A) x = Inf ((\<lambda>X. count X x) ` A)"
   by transfer simp_all
 
 
 instantiation multiset :: (type) Sup
 begin
 
 definition Sup_multiset :: "'a multiset set \<Rightarrow> 'a multiset" where
   "Sup_multiset A = (if A \<noteq> {} \<and> subset_mset.bdd_above A then
            Abs_multiset (\<lambda>i. Sup ((\<lambda>X. count X i) ` A)) else {#})"
 
 lemma Sup_multiset_empty: "Sup {} = {#}"
   by (simp add: Sup_multiset_def)
 
 lemma Sup_multiset_unbounded: "\<not> subset_mset.bdd_above A \<Longrightarrow> Sup A = {#}"
   by (simp add: Sup_multiset_def)
 
 instance ..
 
 end
 
 lemma bdd_above_multiset_imp_bdd_above_count:
   assumes "subset_mset.bdd_above (A :: 'a multiset set)"
   shows   "bdd_above ((\<lambda>X. count X x) ` A)"
 proof -
   from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y"
     by (meson subset_mset.bdd_above.E)
   hence "count X x \<le> count Y x" if "X \<in> A" for X
     using that by (auto intro: mset_subset_eq_count)
   thus ?thesis by (intro bdd_aboveI[of _ "count Y x"]) auto
 qed
 
 lemma bdd_above_multiset_imp_finite_support:
   assumes "A \<noteq> {}" "subset_mset.bdd_above (A :: 'a multiset set)"
   shows   "finite (\<Union>X\<in>A. {x. count X x > 0})"
 proof -
   from assms obtain Y where Y: "\<forall>X\<in>A. X \<subseteq># Y"
     by (meson subset_mset.bdd_above.E)
   hence "count X x \<le> count Y x" if "X \<in> A" for X x
     using that by (auto intro: mset_subset_eq_count)
   hence "(\<Union>X\<in>A. {x. count X x > 0}) \<subseteq> {x. count Y x > 0}"
     by safe (erule less_le_trans)
   moreover have "finite \<dots>" by simp
   ultimately show ?thesis by (rule finite_subset)
 qed
 
 lemma Sup_multiset_in_multiset:
   \<open>finite {i. 0 < (SUP M\<in>A. count M i)}\<close>
   if \<open>A \<noteq> {}\<close> \<open>subset_mset.bdd_above A\<close>
 proof -
   have "{i. Sup ((\<lambda>X. count X i) ` A) > 0} \<subseteq> (\<Union>X\<in>A. {i. 0 < count X i})"
   proof safe
     fix i assume pos: "(SUP X\<in>A. count X i) > 0"
     show "i \<in> (\<Union>X\<in>A. {i. 0 < count X i})"
     proof (rule ccontr)
       assume "i \<notin> (\<Union>X\<in>A. {i. 0 < count X i})"
       hence "\<forall>X\<in>A. count X i \<le> 0" by (auto simp: count_eq_zero_iff)
       with that have "(SUP X\<in>A. count X i) \<le> 0"
         by (intro cSup_least bdd_above_multiset_imp_bdd_above_count) auto
       with pos show False by simp
     qed
   qed
   moreover from that have "finite \<dots>"
     by (rule bdd_above_multiset_imp_finite_support)
   ultimately show "finite {i. Sup ((\<lambda>X. count X i) ` A) > 0}"
     by (rule finite_subset)
 qed
 
 lemma count_Sup_multiset_nonempty:
   \<open>count (Sup A) x = (SUP X\<in>A. count X x)\<close>
   if \<open>A \<noteq> {}\<close> \<open>subset_mset.bdd_above A\<close>
   using that by (simp add: Sup_multiset_def Sup_multiset_in_multiset count_Abs_multiset)
 
 interpretation subset_mset: conditionally_complete_lattice Inf Sup "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)" "(\<union>#)"
 proof
   fix X :: "'a multiset" and A
   assume "X \<in> A"
   show "Inf A \<subseteq># X"
   proof (rule mset_subset_eqI)
     fix x
     from \<open>X \<in> A\<close> have "A \<noteq> {}" by auto
     hence "count (Inf A) x = (INF X\<in>A. count X x)"
       by (simp add: count_Inf_multiset_nonempty)
     also from \<open>X \<in> A\<close> have "\<dots> \<le> count X x"
       by (intro cInf_lower) simp_all
     finally show "count (Inf A) x \<le> count X x" .
   qed
 next
   fix X :: "'a multiset" and A
   assume nonempty: "A \<noteq> {}" and le: "\<And>Y. Y \<in> A \<Longrightarrow> X \<subseteq># Y"
   show "X \<subseteq># Inf A"
   proof (rule mset_subset_eqI)
     fix x
     from nonempty have "count X x \<le> (INF X\<in>A. count X x)"
       by (intro cInf_greatest) (auto intro: mset_subset_eq_count le)
     also from nonempty have "\<dots> = count (Inf A) x" by (simp add: count_Inf_multiset_nonempty)
     finally show "count X x \<le> count (Inf A) x" .
   qed
 next
   fix X :: "'a multiset" and A
   assume X: "X \<in> A" and bdd: "subset_mset.bdd_above A"
   show "X \<subseteq># Sup A"
   proof (rule mset_subset_eqI)
     fix x
     from X have "A \<noteq> {}" by auto
     have "count X x \<le> (SUP X\<in>A. count X x)"
       by (intro cSUP_upper X bdd_above_multiset_imp_bdd_above_count bdd)
     also from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd]
       have "(SUP X\<in>A. count X x) = count (Sup A) x" by simp
     finally show "count X x \<le> count (Sup A) x" .
   qed
 next
   fix X :: "'a multiset" and A
   assume nonempty: "A \<noteq> {}" and ge: "\<And>Y. Y \<in> A \<Longrightarrow> Y \<subseteq># X"
   from ge have bdd: "subset_mset.bdd_above A"
     by blast
   show "Sup A \<subseteq># X"
   proof (rule mset_subset_eqI)
     fix x
     from count_Sup_multiset_nonempty[OF \<open>A \<noteq> {}\<close> bdd]
       have "count (Sup A) x = (SUP X\<in>A. count X x)" .
     also from nonempty have "\<dots> \<le> count X x"
       by (intro cSup_least) (auto intro: mset_subset_eq_count ge)
     finally show "count (Sup A) x \<le> count X x" .
   qed
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 lemma set_mset_Inf:
   assumes "A \<noteq> {}"
   shows   "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)"
 proof safe
   fix x X assume "x \<in># Inf A" "X \<in> A"
   hence nonempty: "A \<noteq> {}" by (auto simp: Inf_multiset_empty)
   from \<open>x \<in># Inf A\<close> have "{#x#} \<subseteq># Inf A" by auto
   also from \<open>X \<in> A\<close> have "\<dots> \<subseteq># X" by (rule subset_mset.cInf_lower) simp_all
   finally show "x \<in># X" by simp
 next
   fix x assume x: "x \<in> (\<Inter>X\<in>A. set_mset X)"
   hence "{#x#} \<subseteq># X" if "X \<in> A" for X using that by auto
   from assms and this have "{#x#} \<subseteq># Inf A" by (rule subset_mset.cInf_greatest)
   thus "x \<in># Inf A" by simp
 qed
 
 lemma in_Inf_multiset_iff:
   assumes "A \<noteq> {}"
   shows   "x \<in># Inf A \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)"
 proof -
   from assms have "set_mset (Inf A) = (\<Inter>X\<in>A. set_mset X)" by (rule set_mset_Inf)
   also have "x \<in> \<dots> \<longleftrightarrow> (\<forall>X\<in>A. x \<in># X)" by simp
   finally show ?thesis .
 qed
 
 lemma in_Inf_multisetD: "x \<in># Inf A \<Longrightarrow> X \<in> A \<Longrightarrow> x \<in># X"
   by (subst (asm) in_Inf_multiset_iff) auto
 
 lemma set_mset_Sup:
   assumes "subset_mset.bdd_above A"
   shows   "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)"
 proof safe
   fix x assume "x \<in># Sup A"
   hence nonempty: "A \<noteq> {}" by (auto simp: Sup_multiset_empty)
   show "x \<in> (\<Union>X\<in>A. set_mset X)"
   proof (rule ccontr)
     assume x: "x \<notin> (\<Union>X\<in>A. set_mset X)"
     have "count X x \<le> count (Sup A) x" if "X \<in> A" for X x
       using that by (intro mset_subset_eq_count subset_mset.cSup_upper assms)
     with x have "X \<subseteq># Sup A - {#x#}" if "X \<in> A" for X
       using that by (auto simp: subseteq_mset_def algebra_simps not_in_iff)
     hence "Sup A \<subseteq># Sup A - {#x#}" by (intro subset_mset.cSup_least nonempty)
     with \<open>x \<in># Sup A\<close> show False
       by (auto simp: subseteq_mset_def simp flip: count_greater_zero_iff
                dest!: spec[of _ x])
   qed
 next
   fix x X assume "x \<in> set_mset X" "X \<in> A"
   hence "{#x#} \<subseteq># X" by auto
   also have "X \<subseteq># Sup A" by (intro subset_mset.cSup_upper \<open>X \<in> A\<close> assms)
   finally show "x \<in> set_mset (Sup A)" by simp
 qed
 
 lemma in_Sup_multiset_iff:
   assumes "subset_mset.bdd_above A"
   shows   "x \<in># Sup A \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)"
 proof -
   from assms have "set_mset (Sup A) = (\<Union>X\<in>A. set_mset X)" by (rule set_mset_Sup)
   also have "x \<in> \<dots> \<longleftrightarrow> (\<exists>X\<in>A. x \<in># X)" by simp
   finally show ?thesis .
 qed
 
 lemma in_Sup_multisetD:
   assumes "x \<in># Sup A"
   shows   "\<exists>X\<in>A. x \<in># X"
 proof -
   have "subset_mset.bdd_above A"
     by (rule ccontr) (insert assms, simp_all add: Sup_multiset_unbounded)
   with assms show ?thesis by (simp add: in_Sup_multiset_iff)
 qed
 
 interpretation subset_mset: distrib_lattice "(\<inter>#)" "(\<subseteq>#)" "(\<subset>#)" "(\<union>#)"
 proof
   fix A B C :: "'a multiset"
   show "A \<union># (B \<inter># C) = A \<union># B \<inter># (A \<union># C)"
     by (intro multiset_eqI) simp_all
 qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
 
 
 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: "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 filter_sup_mset[simp]: "filter_mset P (A \<union># B) = filter_mset P A \<union># filter_mset P B"
   by (rule multiset_eqI) simp
 
 lemma filter_mset_add_mset [simp]:
    "filter_mset P (add_mset x A) =
      (if P x then add_mset x (filter_mset P A) else filter_mset P A)"
    by (auto simp: multiset_eq_iff)
 
 lemma multiset_filter_subset[simp]: "filter_mset f M \<subseteq># M"
   by (simp add: mset_subset_eqI)
 
 lemma multiset_filter_mono:
   assumes "A \<subseteq># B"
   shows "filter_mset f A \<subseteq># filter_mset f B"
 proof -
   from assms[unfolded mset_subset_eq_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_subset_eq_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
 
 lemma filter_filter_mset: "filter_mset P (filter_mset Q M) = {#x \<in># M. Q x \<and> P x#}"
   by (auto simp: multiset_eq_iff)
 
 lemma
   filter_mset_True[simp]: "{#y \<in># M. True#} = M" and
   filter_mset_False[simp]: "{#y \<in># M. False#} = {#}"
   by (auto simp: multiset_eq_iff)
 
+lemma filter_mset_cong0:
+  assumes "\<And>x. x \<in># M \<Longrightarrow> f x \<longleftrightarrow> g x"
+  shows "filter_mset f M = filter_mset g M"
+proof (rule subset_mset.antisym; unfold subseteq_mset_def; rule allI)
+  fix x
+  show "count (filter_mset f M) x \<le> count (filter_mset g M) x"
+    using assms by (cases "x \<in># M") (simp_all add: not_in_iff)
+next
+  fix x
+  show "count (filter_mset g M) x \<le> count (filter_mset f M) x"
+    using assms by (cases "x \<in># M") (simp_all add: not_in_iff)
+qed
+
+lemma filter_mset_cong:
+  assumes "M = M'" and "\<And>x. x \<in># M' \<Longrightarrow> f x \<longleftrightarrow> g x"
+  shows "filter_mset f M = filter_mset g M'"
+  unfolding \<open>M = M'\<close>
+  using assms by (auto intro: filter_mset_cong0)
+
 
 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)
 
 lemma wcount_add_mset:
   "wcount f (add_mset x M) a = (if x = a then Suc (f a) else 0) + wcount f M a"
   unfolding add_mset_add_single[of _ M] wcount_union by (auto simp: wcount_def)
 
 definition size_multiset :: "('a \<Rightarrow> nat) \<Rightarrow> 'a multiset \<Rightarrow> nat" where
   "size_multiset f M = sum (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 : "size_multiset f {#b#} = Suc (f b)"
 by (simp add: size_multiset_eq)
 
 lemma size_single: "size {#b#} = 1"
 by (simp add: size_multiset_overloaded_def size_multiset_single)
 
 lemma sum_wcount_Int:
   "finite A \<Longrightarrow> sum (wcount f N) (A \<inter> set_mset N) = sum (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 sum_Un_nat sum.distrib sum_wcount_Int wcount_union)
 apply (subst Int_commute)
 apply (simp add: sum_wcount_Int)
 done
 
 lemma size_multiset_add_mset [simp]:
   "size_multiset f (add_mset a M) = Suc (f a) + size_multiset f M"
   unfolding add_mset_add_single[of _ M] size_multiset_union by (auto simp: size_multiset_single)
 
 lemma size_add_mset [simp]: "size (add_mset a A) = Suc (size A)"
 by (simp add: size_multiset_overloaded_def wcount_add_mset)
 
 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 sum_SucD)
 apply auto
 done
 
 lemma size_eq_Suc_imp_eq_union:
   assumes "size M = Suc n"
   shows "\<exists>a N. M = add_mset a N"
 proof -
   from assms obtain a where "a \<in># M"
     by (erule size_eq_Suc_imp_elem [THEN exE])
   then have "M = add_mset a (M - {#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_subset_eq_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_subset_eq_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>x M. P M \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case 0 thus "P M" by (simp add: empty)
 next
   case (Suc k)
   obtain N x where "M = add_mset x N"
     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 multiset_induct_min[case_names empty add]:
   fixes M :: "'a::linorder multiset"
   assumes
     empty: "P {#}" and
     add: "\<And>x M. P M \<Longrightarrow> (\<forall>y \<in># M. y \<ge> x) \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case (Suc k)
   note ih = this(1) and Sk_eq_sz_M = this(2)
 
   let ?y = "Min_mset M"
   let ?N = "M - {#?y#}"
 
   have M: "M = add_mset ?y ?N"
     by (metis Min_in Sk_eq_sz_M finite_set_mset insert_DiffM lessI not_less_zero
       set_mset_eq_empty_iff size_empty)
   show ?case
     by (subst M, rule add, rule ih, metis M Sk_eq_sz_M nat.inject size_add_mset,
       meson Min_le finite_set_mset in_diffD)
 qed (simp add: empty)
 
 lemma multiset_induct_max[case_names empty add]:
   fixes M :: "'a::linorder multiset"
   assumes
     empty: "P {#}" and
     add: "\<And>x M. P M \<Longrightarrow> (\<forall>y \<in># M. y \<le> x) \<Longrightarrow> P (add_mset x M)"
   shows "P M"
 proof (induct "size M" arbitrary: M)
   case (Suc k)
   note ih = this(1) and Sk_eq_sz_M = this(2)
 
   let ?y = "Max_mset M"
   let ?N = "M - {#?y#}"
 
   have M: "M = add_mset ?y ?N"
     by (metis Max_in Sk_eq_sz_M finite_set_mset insert_DiffM lessI not_less_zero
       set_mset_eq_empty_iff size_empty)
   show ?case
     by (subst M, rule add, rule ih, metis M Sk_eq_sz_M nat.inject size_add_mset,
       meson Max_ge finite_set_mset in_diffD)
 qed (simp add: empty)
 
 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = add_mset a A"
 by (induct M) auto
 
 lemma multiset_cases [cases type]:
   obtains (empty) "M = {#}"
     | (add) x N where "M = add_mset x N"
   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 union_filter_mset_complement[simp]:
   "\<forall>x. P x = (\<not> Q x) \<Longrightarrow> filter_mset P M + filter_mset Q M = M"
 by (subst multiset_eq_iff) auto
 
 lemma multiset_partition: "M = {#x \<in># M. P x#} + {#x \<in># M. \<not> P x#}"
 by simp
 
 lemma mset_subset_size: "A \<subset># B \<Longrightarrow> size A < size B"
 proof (induct A arbitrary: B)
   case empty
   then show ?case
     using nonempty_has_size by auto
 next
   case (add x A)
   have "add_mset x A \<subseteq># B"
     by (meson add.prems subset_mset_def)
   then show ?case
     by (metis (no_types) add.prems add.right_neutral add_diff_cancel_left' leD nat_neq_iff
       size_Diff_submset size_eq_0_iff_empty size_mset_mono subset_mset.le_iff_add subset_mset_def)
 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_subset_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_subset_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_subset_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 (add_mset a F)"
   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: "add_mset x F \<subseteq># A"
     show "P (add_mset x F)"
     proof (rule insert)
       from i show "x \<in># A" by (auto dest: mset_subset_eq_insertD)
       from i have "F \<subseteq># A" by (auto dest: mset_subset_eq_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_add_mset [simp]: "fold_mset f s (add_mset x M) = 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 (add_mset x M) y"
     by (fact comp_fun_commute_funpow)
   show ?thesis
   proof (cases "x \<in> set_mset M")
     case False
     then have *: "count (add_mset x M) x = 1"
       by (simp add: not_in_iff)
     from False have "Finite_Set.fold (\<lambda>y. f y ^^ count (add_mset x M) 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_on_funpow)
     with False * show ?thesis
       by (simp add: fold_mset_def del: count_add_mset)
   next
     case True
     define N where "N = 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 (add_mset x M) 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_on_funpow)
     with * show ?thesis by (simp add: fold_mset_def del: count_add_mset) simp
   qed
 qed
 
 corollary fold_mset_single: "fold_mset f s {#x#} = f x s"
   by simp
 
 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: fun_left_comm)
 
 lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
   by (induct M) (simp_all add: fold_mset_fun_left_comm)
 
 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
 
 lemma union_fold_mset_add_mset: "A + B = fold_mset add_mset A B"
 proof -
   interpret comp_fun_commute add_mset
     by standard auto
   show ?thesis
     by (induction B) auto
 qed
 
 text \<open>
   A note on code generation: When defining some function containing a
   subterm \<^term>\<open>fold_mset F\<close>, code generation is not automatic. When
   interpreting locale \<open>left_commutative\<close> with \<open>F\<close>, the
   would be code thms for \<^const>\<open>fold_mset\<close> become thms like
   \<^term>\<open>fold_mset F z {#} = z\<close> 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 (add_mset \<circ> f) {#}"
 
 lemma comp_fun_commute_mset_image: "comp_fun_commute (add_mset \<circ> f)"
   by unfold_locales (simp add: fun_eq_iff)
 
 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
   by (simp add: image_mset_def)
 
 lemma image_mset_single: "image_mset f {#x#} = {#f x#}"
   by (simp add: comp_fun_commute.fold_mset_add_mset comp_fun_commute_mset_image image_mset_def)
 
 lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
 proof -
   interpret comp_fun_commute "add_mset \<circ> f"
     by (fact comp_fun_commute_mset_image)
   show ?thesis by (induct N) (simp_all add: image_mset_def)
 qed
 
 corollary image_mset_add_mset [simp]:
   "image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"
   unfolding image_mset_union add_mset_add_single[of a M] by (simp add: image_mset_single)
 
 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
 
 lemma image_mset_If:
   "image_mset (\<lambda>x. if P x then f x else g x) A =
      image_mset f (filter_mset P A) + image_mset g (filter_mset (\<lambda>x. \<not>P x) A)"
   by (induction A) auto
 
 lemma image_mset_Diff:
   assumes "B \<subseteq># A"
   shows   "image_mset f (A - B) = image_mset f A - image_mset f B"
 proof -
   have "image_mset f (A - B + B) = image_mset f (A - B) + image_mset f B"
     by simp
   also from assms have "A - B + B = A"
     by (simp add: subset_mset.diff_add)
   finally show ?thesis by simp
 qed
 
 lemma count_image_mset:
   \<open>count (image_mset f A) x = (\<Sum>y\<in>f -` {x} \<inter> set_mset A. count A y)\<close>
 proof (induction A)
   case empty
   then show ?case by simp
 next
   case (add x A)
   moreover have *: "(if x = y then Suc n else n) = n + (if x = y then 1 else 0)" for n y
     by simp
   ultimately show ?case
     by (auto simp: sum.distrib intro!: sum.mono_neutral_left)
 qed
 
 lemma count_image_mset':
   \<open>count (image_mset f X) y = (\<Sum>x | x \<in># X \<and> y = f x. count X x)\<close>
   by (auto simp add: count_image_mset simp flip: singleton_conv2 simp add: Collect_conj_eq ac_simps)
 
 lemma image_mset_subseteq_mono: "A \<subseteq># B \<Longrightarrow> image_mset f A \<subseteq># image_mset f B"
   by (metis image_mset_union subset_mset.le_iff_add)
 
 lemma image_mset_subset_mono: "M \<subset># N \<Longrightarrow> image_mset f M \<subset># image_mset f N"
   by (metis (no_types) Diff_eq_empty_iff_mset image_mset_Diff image_mset_is_empty_iff
     image_mset_subseteq_mono subset_mset.less_le_not_le)
 
 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>\<open>{#x\<in>#M. x<c#}\<close>
   but also images like \<^term>\<open>{#x+x. x\<in>#M #}\<close> and @{term [source]
   "{#x+x|x\<in>#M. x<c#}"}, where the latter is currently displayed as
   \<^term>\<open>{#x+x|x\<in>#M. x<c#}\<close>.
 \<close>
 
 lemma in_image_mset: "y \<in># {#f x. x \<in># M#} \<longleftrightarrow> y \<in> f ` set_mset M"
   by simp
 
 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)
 
 lemma image_mset_const_eq:
   "{#c. a \<in># M#} = replicate_mset (size M) c"
   by (induct M) simp_all
 
 
 subsection \<open>Further conversions\<close>
 
 primrec mset :: "'a list \<Rightarrow> 'a multiset" where
   "mset [] = {#}" |
   "mset (a # x) = add_mset a (mset x)"
 
 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 count_mset_gt_0: "x \<in> set xs \<Longrightarrow> count (mset xs) x > 0"
   by (induction xs) auto
 
 lemma count_mset_0_iff [simp]: "count (mset xs) x = 0 \<longleftrightarrow> x \<notin> set xs"
   by (induction xs) auto
 
 lemma mset_single_iff[iff]: "mset xs = {#x#} \<longleftrightarrow> xs = [x]"
   by (cases xs) auto
 
 lemma mset_single_iff_right[iff]: "{#x#} = mset xs \<longleftrightarrow> xs = [x]"
   by (cases xs) auto
 
 lemma set_mset_mset[simp]: "set_mset (mset xs) = set xs"
   by (induct xs) 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
 
 lemma mset_filter[simp]: "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:
   \<open>distinct x \<Longrightarrow> distinct y \<Longrightarrow> set x = set y \<longleftrightarrow> mset x = mset y\<close>
   by (auto simp: multiset_eq_iff distinct_count_atmost_1)
 
 lemma set_eq_iff_mset_remdups_eq:
   \<open>set x = set y \<longleftrightarrow> mset (remdups x) = mset (remdups y)\<close>
 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_eq_imp_distinct_iff:
   \<open>distinct xs \<longleftrightarrow> distinct ys\<close> if \<open>mset xs = mset ys\<close>
   using that by (auto simp add: distinct_count_atmost_1 dest: mset_eq_setD)
 
 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:
   \<open>List.fold f xs = List.fold f ys\<close>
     if f: \<open>\<And>x y. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> f x \<circ> f y = f y \<circ> f x\<close>
     and \<open>mset xs = mset ys\<close>
 using f \<open>mset xs = mset ys\<close> [symmetric] proof (induction xs arbitrary: ys)
   case Nil
   then show ?case by simp
 next
   case (Cons x xs)
   then have *: \<open>set ys = set (x # xs)\<close>
     by (blast dest: mset_eq_setD)
   have \<open>\<And>x y. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> f x \<circ> f y = f y \<circ> f x\<close>
     by (rule Cons.prems(1)) (simp_all add: *)
   moreover from * have \<open>x \<in> set ys\<close>
     by simp
   ultimately have \<open>List.fold f ys = List.fold f (remove1 x ys) \<circ> f x\<close>
     by (fact fold_remove1_split)
   moreover from Cons.prems have \<open>List.fold f xs = List.fold f (remove1 x ys)\<close>
     by (auto intro: Cons.IH)
   ultimately show ?case
     by simp
 qed
 
 lemma fold_permuted_eq:
   \<open>List.fold (\<odot>) xs z = List.fold (\<odot>) ys z\<close>
     if \<open>mset xs = mset ys\<close>
     and \<open>P z\<close> and P: \<open>\<And>x z. x \<in> set xs \<Longrightarrow> P z \<Longrightarrow> P (x \<odot> z)\<close>
     and f: \<open>\<And>x y z. x \<in> set xs \<Longrightarrow> y \<in> set xs \<Longrightarrow> P z \<Longrightarrow> x \<odot> (y \<odot> z) = y \<odot> (x \<odot> z)\<close>
   for f (infixl \<open>\<odot>\<close> 70)
 using \<open>P z\<close> P f \<open>mset xs = mset ys\<close> [symmetric] proof (induction xs arbitrary: ys z)
   case Nil
   then show ?case by simp
 next
   case (Cons x xs)
   then have *: \<open>set ys = set (x # xs)\<close>
     by (blast dest: mset_eq_setD)
   have \<open>P z\<close>
     by (fact Cons.prems(1))
   moreover have \<open>\<And>x z. x \<in> set ys \<Longrightarrow> P z \<Longrightarrow> P (x \<odot> z)\<close>
     by (rule Cons.prems(2)) (simp_all add: *)
   moreover have \<open>\<And>x y z. x \<in> set ys \<Longrightarrow> y \<in> set ys \<Longrightarrow> P z \<Longrightarrow> x \<odot> (y \<odot> z) = y \<odot> (x \<odot> z)\<close>
     by (rule Cons.prems(3)) (simp_all add: *)
   moreover from * have \<open>x \<in> set ys\<close>
     by simp
   ultimately have \<open>fold (\<odot>) ys z = fold (\<odot>) (remove1 x ys) (x \<odot> z)\<close>
     by (induction ys arbitrary: z) auto
   moreover from Cons.prems have \<open>fold (\<odot>) xs (x \<odot> z) = fold (\<odot>) (remove1 x ys) (x \<odot> z)\<close>
     by (auto intro: Cons.IH)
   ultimately show ?case
     by simp
 qed
 
 lemma mset_shuffles: "zs \<in> shuffles xs ys \<Longrightarrow> mset zs = mset xs + mset ys"
   by (induction xs ys arbitrary: zs rule: shuffles.induct) auto
 
 lemma mset_insort [simp]: "mset (insort x xs) = add_mset x (mset xs)"
   by (induct xs) simp_all
 
 lemma mset_map[simp]: "mset (map f xs) = image_mset f (mset xs)"
   by (induct xs) simp_all
 
 global_interpretation mset_set: folding add_mset "{#}"
   defines mset_set = "folding_on.F add_mset {#}"
   by standard (simp add: fun_eq_iff)
 
 lemma sum_multiset_singleton [simp]: "sum (\<lambda>n. {#n#}) A = mset_set A"
   by (induction A rule: infinite_finite_induct) auto
 
 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>\<open>mset_set\<close> also in terms of \<^const>\<open>Abs_multiset\<close>\<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
 
 lemma mset_set_Union:
   "finite A \<Longrightarrow> finite B \<Longrightarrow> A \<inter> B = {} \<Longrightarrow> mset_set (A \<union> B) = mset_set A + mset_set B"
   by (induction A rule: finite_induct) auto
 
 lemma filter_mset_mset_set [simp]:
   "finite A \<Longrightarrow> filter_mset P (mset_set A) = mset_set {x\<in>A. P x}"
 proof (induction A rule: finite_induct)
   case (insert x A)
   from insert.hyps have "filter_mset P (mset_set (insert x A)) =
       filter_mset P (mset_set A) + mset_set (if P x then {x} else {})"
     by simp
   also have "filter_mset P (mset_set A) = mset_set {x\<in>A. P x}"
     by (rule insert.IH)
   also from insert.hyps
     have "\<dots> + mset_set (if P x then {x} else {}) =
             mset_set ({x \<in> A. P x} \<union> (if P x then {x} else {}))" (is "_ = mset_set ?A")
      by (intro mset_set_Union [symmetric]) simp_all
   also from insert.hyps have "?A = {y\<in>insert x A. P y}" by auto
   finally show ?case .
 qed simp_all
 
 lemma mset_set_Diff:
   assumes "finite A" "B \<subseteq> A"
   shows  "mset_set (A - B) = mset_set A - mset_set B"
 proof -
   from assms have "mset_set ((A - B) \<union> B) = mset_set (A - B) + mset_set B"
     by (intro mset_set_Union) (auto dest: finite_subset)
   also from assms have "A - B \<union> B = A" by blast
   finally show ?thesis by simp
 qed
 
 lemma mset_set_set: "distinct xs \<Longrightarrow> mset_set (set xs) = mset xs"
   by (induction xs) simp_all
 
 lemma count_mset_set': "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
   by auto
 
 lemma subset_imp_msubset_mset_set: 
   assumes "A \<subseteq> B" "finite B"
   shows   "mset_set A \<subseteq># mset_set B"
 proof (rule mset_subset_eqI)
   fix x :: 'a
   from assms have "finite A" by (rule finite_subset)
   with assms show "count (mset_set A) x \<le> count (mset_set B) x"
     by (cases "x \<in> A"; cases "x \<in> B") auto
 qed
 
 lemma mset_set_set_mset_msubset: "mset_set (set_mset A) \<subseteq># A"
 proof (rule mset_subset_eqI)
   fix x show "count (mset_set (set_mset A)) x \<le> count A x"
     by (cases "x \<in># A") simp_all
 qed
 
 lemma mset_set_upto_eq_mset_upto:
   \<open>mset_set {..<n} = mset [0..<n]\<close>
   by (induction n) (auto simp: ac_simps lessThan_Suc)
 
 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 (add_mset x M) = 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 auto
 
 lemma mset_set_empty_iff: "mset_set A = {#} \<longleftrightarrow> A = {} \<or> infinite A"
   using finite_set_mset_mset_set by fastforce
 
 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_key)
 
 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)
 
 lemma mset_upt [simp]: "mset [m..<n] = mset_set {m..<n}"
   by (induction n) (simp_all add: atLeastLessThanSuc)
 
 lemma image_mset_map_of:
   "distinct (map fst xs) \<Longrightarrow> {#the (map_of xs i). i \<in># mset (map fst xs)#} = mset (map snd xs)"
 proof (induction xs)
   case (Cons x xs)
   have "{#the (map_of (x # xs) i). i \<in># mset (map fst (x # xs))#} =
           add_mset (snd x) {#the (if i = fst x then Some (snd x) else map_of xs i).
              i \<in># mset (map fst xs)#}" (is "_ = add_mset _ ?A") by simp
   also from Cons.prems have "?A = {#the (map_of xs i). i :# mset (map fst xs)#}"
     by (cases x, intro image_mset_cong) (auto simp: in_multiset_in_set)
   also from Cons.prems have "\<dots> = mset (map snd xs)" by (intro Cons.IH) simp_all
   finally show ?case by simp
 qed simp_all
 
 lemma msubset_mset_set_iff[simp]:
   assumes "finite A" "finite B"
   shows "mset_set A \<subseteq># mset_set B \<longleftrightarrow> A \<subseteq> B"
   using assms set_mset_mono subset_imp_msubset_mset_set by fastforce
 
 lemma mset_set_eq_iff[simp]:
   assumes "finite A" "finite B"
   shows "mset_set A = mset_set B \<longleftrightarrow> A = B"
   using assms by (fastforce dest: finite_set_mset_mset_set)
 
 lemma image_mset_mset_set: \<^marker>\<open>contributor \<open>Lukas Bulwahn\<close>\<close>
   assumes "inj_on f A"
   shows "image_mset f (mset_set A) = mset_set (f ` A)"
 proof cases
   assume "finite A"
   from this \<open>inj_on f A\<close> show ?thesis
     by (induct A) auto
 next
   assume "infinite A"
   from this \<open>inj_on f A\<close> have "infinite (f ` A)"
     using finite_imageD by blast
   from \<open>infinite A\<close> \<open>infinite (f ` A)\<close> show ?thesis by simp
 qed
 
 
 subsection \<open>More properties of the replicate and repeat operations\<close>
 
 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 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_subset_eq: "n \<le> count M x \<longleftrightarrow> replicate_mset n x \<subseteq># M"
   by (auto simp add: mset_subset_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)
 
 lemma repeat_mset_cancel1: "repeat_mset a A = repeat_mset a B \<longleftrightarrow> A = B \<or> a = 0"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_cancel2: "repeat_mset a A = repeat_mset b A \<longleftrightarrow> a = b \<or> A = {#}"
   by (auto simp: multiset_eq_iff)
 
 lemma repeat_mset_eq_empty_iff: "repeat_mset n A = {#} \<longleftrightarrow> n = 0 \<or> A = {#}"
   by (cases n) auto
 
 lemma image_replicate_mset [simp]:
   "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
   by (induct n) simp_all
 
 lemma replicate_mset_msubseteq_iff:
   "replicate_mset m a \<subseteq># replicate_mset n b \<longleftrightarrow> m = 0 \<or> a = b \<and> m \<le> n"
   by (cases m)
     (auto simp: insert_subset_eq_iff simp flip: count_le_replicate_mset_subset_eq)
 
 lemma msubseteq_replicate_msetE:
   assumes "A \<subseteq># replicate_mset n a"
   obtains m where "m \<le> n" and "A = replicate_mset m a"
 proof (cases "n = 0")
   case True
   with assms that show thesis
     by simp
 next
   case False
   from assms have "set_mset A \<subseteq> set_mset (replicate_mset n a)"
     by (rule set_mset_mono)
   with False have "set_mset A \<subseteq> {a}"
     by simp
   then have "\<exists>m. A = replicate_mset m a"
   proof (induction A)
     case empty
     then show ?case
       by simp
   next
     case (add b A)
     then obtain m where "A = replicate_mset m a"
       by auto
     with add.prems show ?case
       by (auto intro: exI [of _ "Suc m"])
   qed
   then obtain m where A: "A = replicate_mset m a" ..
   with assms have "m \<le> n"
     by (auto simp add: replicate_mset_msubseteq_iff)
   then show thesis using A ..
 qed
 
 
 subsection \<open>Big operators\<close>
 
 locale comm_monoid_mset = comm_monoid
 begin
 
 interpretation comp_fun_commute f
   by standard (simp add: fun_eq_iff left_commute)
 
 interpretation comp?: comp_fun_commute "f \<circ> g"
   by (fact comp_comp_fun_commute)
 
 context
 begin
 
 definition F :: "'a multiset \<Rightarrow> 'a"
   where eq_fold: "F M = fold_mset f \<^bold>1 M"
 
 lemma empty [simp]: "F {#} = \<^bold>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 \<^bold>* 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
 
 lemma add_mset [simp]: "F (add_mset x N) = x \<^bold>* F N"
   unfolding add_mset_add_single[of x N] union by (simp add: ac_simps)
 
 lemma insert [simp]:
   shows "F (image_mset g (add_mset x A)) = g x \<^bold>* F (image_mset g A)"
   by (simp add: eq_fold)
 
 lemma remove:
   assumes "x \<in># A"
   shows "F A = x \<^bold>* F (A - {#x#})"
   using multi_member_split[OF assms] by auto
 
 lemma neutral:
   "\<forall>x\<in>#A. x = \<^bold>1 \<Longrightarrow> F A = \<^bold>1"
   by (induct A) simp_all
 
 lemma neutral_const [simp]:
   "F (image_mset (\<lambda>_. \<^bold>1) A) = \<^bold>1"
   by (simp add: neutral)
 
 private lemma F_image_mset_product:
   "F {#g x j \<^bold>* F {#g i j. i \<in># A#}. j \<in># B#} =
     F (image_mset (g x) B) \<^bold>* F {#F {#g i j. i \<in># A#}. j \<in># B#}"
   by (induction B) (simp_all add: left_commute semigroup.assoc semigroup_axioms)
 
 lemma swap:
   "F (image_mset (\<lambda>i. F (image_mset (g i) B)) A) =
     F (image_mset (\<lambda>j. F (image_mset (\<lambda>i. g i j) A)) B)"
   apply (induction A, simp)
   apply (induction B, auto simp add: F_image_mset_product ac_simps)
   done
 
 lemma distrib: "F (image_mset (\<lambda>x. g x \<^bold>* h x) A) = F (image_mset g A) \<^bold>* F (image_mset h A)"
   by (induction A) (auto simp: ac_simps)
 
 lemma union_disjoint:
   "A \<inter># B = {#} \<Longrightarrow> F (image_mset g (A \<union># B)) = F (image_mset g A) \<^bold>* F (image_mset g B)"
   by (induction A) (auto simp: ac_simps)
 
 end
 end
 
 lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute ((+) :: 'a multiset \<Rightarrow> _ \<Rightarrow> _)"
   by standard (simp add: add_ac comp_def)
 
 declare comp_fun_commute.fold_mset_add_mset[OF comp_fun_commute_plus_mset, simp]
 
 lemma in_mset_fold_plus_iff[iff]: "x \<in># fold_mset (+) M NN \<longleftrightarrow> x \<in># M \<or> (\<exists>N. N \<in># NN \<and> x \<in># N)"
   by (induct NN) auto
 
 context comm_monoid_add
 begin
 
 sublocale sum_mset: comm_monoid_mset plus 0
   defines sum_mset = sum_mset.F ..
 
 lemma sum_unfold_sum_mset:
   "sum f A = sum_mset (image_mset f (mset_set A))"
   by (cases "finite A") (induct A rule: finite_induct, simp_all)
 
 end
 
 notation sum_mset ("\<Sum>\<^sub>#")
 
 syntax (ASCII)
   "_sum_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
 syntax
   "_sum_mset_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 sum_mset (CONST image_mset (\<lambda>i. b) A)"
 
 context comm_monoid_add
 begin
 
 lemma sum_mset_sum_list:
   "sum_mset (mset xs) = sum_list xs"
   by (induction xs) auto
 
 end
 
 context canonically_ordered_monoid_add
 begin
 
 lemma sum_mset_0_iff [simp]:
   "sum_mset M = 0  \<longleftrightarrow> (\<forall>x \<in> set_mset M. x = 0)"
   by (induction M) auto
 
 end
 
 context ordered_comm_monoid_add
 begin
 
 lemma sum_mset_mono:
   "sum_mset (image_mset f K) \<le> sum_mset (image_mset g K)"
   if "\<And>i. i \<in># K \<Longrightarrow> f i \<le> g i"
   using that by (induction K) (simp_all add: add_mono)
 
 end
 
 context cancel_comm_monoid_add
 begin
 
 lemma sum_mset_diff:
   "sum_mset (M - N) = sum_mset M - sum_mset N" if "N \<subseteq># M" for M N :: "'a multiset"
   using that by (auto simp add: subset_mset.le_iff_add)
 
 end
 
 context semiring_0
 begin
 
 lemma sum_mset_distrib_left:
   "c * (\<Sum>x \<in># M. f x) = (\<Sum>x \<in># M. c * f(x))"
   by (induction M) (simp_all add: algebra_simps)
 
 lemma sum_mset_distrib_right:
   "(\<Sum>x \<in># M. f x) * c = (\<Sum>x \<in># M. f x * c)"
   by (induction M) (simp_all add: algebra_simps)
 
 end
 
 lemma sum_mset_product:
   fixes f :: "'a::{comm_monoid_add,times} \<Rightarrow> 'b::semiring_0"
   shows "(\<Sum>i \<in># A. f i) * (\<Sum>i \<in># B. g i) = (\<Sum>i\<in>#A. \<Sum>j\<in>#B. f i * g j)"
   by (subst sum_mset.swap) (simp add: sum_mset_distrib_left sum_mset_distrib_right)
 
 context semiring_1
 begin
 
 lemma sum_mset_replicate_mset [simp]:
   "sum_mset (replicate_mset n a) = of_nat n * a"
   by (induction n) (simp_all add: algebra_simps)
 
 lemma sum_mset_delta:
   "sum_mset (image_mset (\<lambda>x. if x = y then c else 0) A) = c * of_nat (count A y)"
   by (induction A) (simp_all add: algebra_simps)
 
 lemma sum_mset_delta':
   "sum_mset (image_mset (\<lambda>x. if y = x then c else 0) A) = c * of_nat (count A y)"
   by (induction A) (simp_all add: algebra_simps)
 
 end
 
 lemma of_nat_sum_mset [simp]:
   "of_nat (sum_mset A) = sum_mset (image_mset of_nat A)"
   by (induction A) auto
 
 lemma size_eq_sum_mset:
   "size M = (\<Sum>a\<in>#M. 1)"
   using image_mset_const_eq [of "1::nat" M] by simp
 
 lemma size_mset_set [simp]:
   "size (mset_set A) = card A"
   by (simp only: size_eq_sum_mset card_eq_sum sum_unfold_sum_mset)
 
 lemma sum_mset_constant [simp]:
   fixes y :: "'b::semiring_1"
   shows \<open>(\<Sum>x\<in>#A. y) = of_nat (size A) * y\<close>
   by (induction A) (auto simp: algebra_simps)
 
 lemma set_mset_Union_mset[simp]: "set_mset (\<Sum>\<^sub># MM) = (\<Union>M \<in> set_mset MM. set_mset M)"
   by (induct MM) auto
 
 lemma in_Union_mset_iff[iff]: "x \<in># \<Sum>\<^sub># MM \<longleftrightarrow> (\<exists>M. M \<in># MM \<and> x \<in># M)"
   by (induct MM) auto
 
 lemma count_sum:
   "count (sum f A) x = sum (\<lambda>a. count (f a) x) A"
   by (induct A rule: infinite_finite_induct) simp_all
 
 lemma sum_eq_empty_iff:
   assumes "finite A"
   shows "sum f A = {#} \<longleftrightarrow> (\<forall>a\<in>A. f a = {#})"
   using assms by induct simp_all
 
 lemma Union_mset_empty_conv[simp]: "\<Sum>\<^sub># M = {#} \<longleftrightarrow> (\<forall>i\<in>#M. i = {#})"
   by (induction M) auto
 
 lemma Union_image_single_mset[simp]: "\<Sum>\<^sub># (image_mset (\<lambda>x. {#x#}) m) = m"
 by(induction m) auto
 
 
 context comm_monoid_mult
 begin
 
 sublocale prod_mset: comm_monoid_mset times 1
   defines prod_mset = prod_mset.F ..
 
 lemma prod_mset_empty:
   "prod_mset {#} = 1"
   by (fact prod_mset.empty)
 
 lemma prod_mset_singleton:
   "prod_mset {#x#} = x"
   by (fact prod_mset.singleton)
 
 lemma prod_mset_Un:
   "prod_mset (A + B) = prod_mset A * prod_mset B"
   by (fact prod_mset.union)
 
 lemma prod_mset_prod_list:
   "prod_mset (mset xs) = prod_list xs"
   by (induct xs) auto
 
 lemma prod_mset_replicate_mset [simp]:
   "prod_mset (replicate_mset n a) = a ^ n"
   by (induct n) simp_all
 
 lemma prod_unfold_prod_mset:
   "prod f A = prod_mset (image_mset f (mset_set A))"
   by (cases "finite A") (induct A rule: finite_induct, simp_all)
 
 lemma prod_mset_multiplicity:
   "prod_mset M = prod (\<lambda>x. x ^ count M x) (set_mset M)"
   by (simp add: fold_mset_def prod.eq_fold prod_mset.eq_fold funpow_times_power comp_def)
 
 lemma prod_mset_delta: "prod_mset (image_mset (\<lambda>x. if x = y then c else 1) A) = c ^ count A y"
   by (induction A) simp_all
 
 lemma prod_mset_delta': "prod_mset (image_mset (\<lambda>x. if y = x then c else 1) A) = c ^ count A y"
   by (induction A) simp_all
 
 lemma prod_mset_subset_imp_dvd:
   assumes "A \<subseteq># B"
   shows   "prod_mset A dvd prod_mset B"
 proof -
   from assms have "B = (B - A) + A" by (simp add: subset_mset.diff_add)
   also have "prod_mset \<dots> = prod_mset (B - A) * prod_mset A" by simp
   also have "prod_mset A dvd \<dots>" by simp
   finally show ?thesis .
 qed
 
 lemma dvd_prod_mset:
   assumes "x \<in># A"
   shows "x dvd prod_mset A"
   using assms prod_mset_subset_imp_dvd [of "{#x#}" A] by simp
 
 end
 
 notation prod_mset ("\<Prod>\<^sub>#")
 
 syntax (ASCII)
   "_prod_mset_image" :: "pttrn \<Rightarrow> 'b set \<Rightarrow> 'a \<Rightarrow> 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
 syntax
   "_prod_mset_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 prod_mset (CONST image_mset (\<lambda>i. b) A)"
 
 lemma prod_mset_constant [simp]: "(\<Prod>_\<in>#A. c) = c ^ size A"
   by (simp add: image_mset_const_eq)
 
 lemma (in semidom) prod_mset_zero_iff [iff]:
   "prod_mset A = 0 \<longleftrightarrow> 0 \<in># A"
   by (induct A) auto
 
 lemma (in semidom_divide) prod_mset_diff:
   assumes "B \<subseteq># A" and "0 \<notin># B"
   shows "prod_mset (A - B) = prod_mset A div prod_mset 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) prod_mset_minus:
   assumes "a \<in># A" and "a \<noteq> 0"
   shows "prod_mset (A - {#a#}) = prod_mset A div a"
   using assms prod_mset_diff [of "{#a#}" A] by auto
 
 lemma (in normalization_semidom) normalize_prod_mset_normalize:
   "normalize (prod_mset (image_mset normalize A)) = normalize (prod_mset A)"
 proof (induction A)
   case (add x A)
   have "normalize (prod_mset (image_mset normalize (add_mset x A))) =
           normalize (x * normalize (prod_mset (image_mset normalize A)))"
     by simp
   also note add.IH
   finally show ?case by simp
 qed auto
 
 lemma (in algebraic_semidom) is_unit_prod_mset_iff:
   "is_unit (prod_mset A) \<longleftrightarrow> (\<forall>x \<in># A. is_unit x)"
   by (induct A) (auto simp: is_unit_mult_iff)
 
 lemma (in normalization_semidom_multiplicative) normalize_prod_mset:
   "normalize (prod_mset A) = prod_mset (image_mset normalize A)"
   by (induct A) (simp_all add: normalize_mult)
 
 lemma (in normalization_semidom_multiplicative) normalized_prod_msetI:
   assumes "\<And>a. a \<in># A \<Longrightarrow> normalize a = a"
   shows "normalize (prod_mset A) = prod_mset A"
 proof -
   from assms have "image_mset normalize A = A"
     by (induct A) simp_all
   then show ?thesis by (simp add: normalize_prod_mset)
 qed
 
 
 subsection \<open>Multiset as order-ignorant lists\<close>
 
 context linorder
 begin
 
 lemma mset_insort [simp]:
   "mset (insort_key k x xs) = add_mset x (mset xs)"
   by (induct xs) simp_all
 
 lemma mset_sort [simp]:
   "mset (sort_key k xs) = mset xs"
   by (induct xs) simp_all
 
 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
   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 "(<)"] this [of "(>)"] this [of "(=)"]
   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 parameterized 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_subset_eq: "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]) = add_mset v (mset ls - {#ls ! i#})"
 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
       by (cases \<open>x = xs ! i'\<close>) auto
   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)
 
 lemma mset_eq_finite:
   \<open>finite {ys. mset ys = mset xs}\<close>
 proof -
   have \<open>{ys. mset ys = mset xs} \<subseteq> {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
     by (auto simp add: dest: mset_eq_setD mset_eq_length)
   moreover have \<open>finite {ys. set ys \<subseteq> set xs \<and> length ys \<le> length xs}\<close>
     using finite_lists_length_le by blast
   ultimately show ?thesis
     by (rule finite_subset)
 qed
 
 
 subsection \<open>The multiset order\<close>
 
 definition mult1 :: "('a \<times> 'a) set \<Rightarrow> ('a multiset \<times> 'a multiset) set" where
   "mult1 r = {(N, M). \<exists>a M0 K. M = add_mset a M0 \<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>+"
 
 definition multp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multp r M N \<longleftrightarrow> (M, N) \<in> mult {(x, y). r x y}"
 
 lemma mult1I:
   assumes "M = add_mset a M0" 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 = add_mset a M0" "N = M0 + K" "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
   using assms unfolding mult1_def by blast
 
 lemma mono_mult1:
   assumes "r \<subseteq> r'" shows "mult1 r \<subseteq> mult1 r'"
   unfolding mult1_def using assms by blast
 
 lemma mono_mult:
   assumes "r \<subseteq> r'" shows "mult r \<subseteq> mult r'"
   unfolding mult_def using mono_mult1[OF assms] trancl_mono by blast
 
 lemma mono_multp[mono]: "r \<le> r' \<Longrightarrow> multp r \<le> multp r'"
   unfolding le_fun_def le_bool_def
 proof (intro allI impI)
   fix M N :: "'a multiset"
   assume "\<forall>x xa. r x xa \<longrightarrow> r' x xa"
   hence "{(x, y). r x y} \<subseteq> {(x, y). r' x y}"
     by blast
   thus "multp r M N \<Longrightarrow> multp r' M N"
     unfolding multp_def
     by (fact mono_mult[THEN subsetD, rotated])
 qed
 
 lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
   by (simp add: mult1_def)
 
 
 subsubsection \<open>Well-foundedness\<close>
 
 lemma less_add:
   assumes mult1: "(N, add_mset a M0) \<in> mult1 r"
   shows
     "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = add_mset a M) \<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 = add_mset a M0 \<and> N = M0 + K \<and> ?r K a"
   obtain a' M0' K where M0: "add_mset a M0 = add_mset a' M0'"
     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 = add_mset a' K'" "M0' = add_mset a K'"
       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 = add_mset a (K' + K)" by simp
       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. add_mset b M \<in> ?W)"
       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R \<longrightarrow> add_mset a M \<in> ?W"
     have "add_mset a M0 \<in> ?W"
     proof (rule accI [of "add_mset a M0"])
       fix N
       assume "(N, add_mset a M0) \<in> ?R"
       then consider M where "(M, M0) \<in> ?R" "N = add_mset a M"
         | 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> add_mset a M \<in> ?W" ..
         from this and \<open>(M, M0) \<in> ?R\<close> have "add_mset a M \<in> ?W" ..
         then show "N \<in> ?W" by (simp only: \<open>N = add_mset a M\<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 x K)
           from add.prems have "(x, a) \<in> r" by simp
           with wf_hyp have "\<forall>M \<in> ?W. add_mset x M \<in> ?W" by blast
           moreover from add have "M0 + K \<in> ?W" by simp
           ultimately have "add_mset x (M0 + K) \<in> ?W" ..
           then show "M0 + (add_mset x K) \<in> ?W" by simp
         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. add_mset a M \<in> ?W"
     proof induct
       fix a
       assume r: "\<And>b. (b, a) \<in> r \<Longrightarrow> (\<forall>M \<in> ?W. add_mset b M \<in> ?W)"
       show "\<forall>M \<in> ?W. add_mset a M \<in> ?W"
       proof
         fix M assume "M \<in> ?W"
         then show "add_mset a M \<in> ?W"
           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
       qed
     qed
     from this and \<open>M \<in> ?W\<close> show "add_mset a M \<in> ?W" ..
   qed
 qed
 
 lemma wf_mult1: "wf r \<Longrightarrow> wf (mult1 r)"
   by (rule acc_wfI) (rule all_accessible)
 
 lemma wf_mult: "wf r \<Longrightarrow> wf (mult r)"
   unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
 
 lemma wfP_multp: "wfP r \<Longrightarrow> wfP (multp r)"
   unfolding multp_def wfP_def
   by (simp add: wf_mult)
 
 
 subsubsection \<open>Closure-free presentation\<close>
 
 text \<open>One direction.\<close>
 lemma mult_implies_one_step:
   assumes
     trans: "trans r" and
     MN: "(M, N) \<in> mult r"
   shows "\<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)"
   using MN unfolding mult_def mult1_def
 proof (induction rule: converse_trancl_induct)
   case (base y)
   then show ?case by force
 next
   case (step y z) note yz = this(1) and zN = this(2) and N_decomp = this(3)
   obtain I J K where
     N: "N = I + J" "z = I + K" "J \<noteq> {#}" "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r"
     using N_decomp by blast
   obtain a M0 K' where
     z: "z = add_mset a M0" and y: "y = M0 + K'" and K: "\<forall>b. b \<in># K' \<longrightarrow> (b, a) \<in> r"
     using yz by blast
   show ?case
   proof (cases "a \<in># K")
     case True
     moreover have "\<exists>j\<in>#J. (k, j) \<in> r" if "k \<in># K'" for k
       using K N trans True by (meson that transE)
     ultimately show ?thesis
       by (rule_tac x = I in exI, rule_tac x = J in exI, rule_tac x = "(K - {#a#}) + K'" in exI)
         (use z y N in \<open>auto simp del: subset_mset.add_diff_assoc2 dest: in_diffD\<close>)
   next
     case False
     then have "a \<in># I" by (metis N(2) union_iff union_single_eq_member z)
     moreover have "M0 = I + K - {#a#}"
       using N(2) z by force
     ultimately show ?thesis
       by (rule_tac x = "I - {#a#}" in exI, rule_tac x = "add_mset a J" in exI,
           rule_tac x = "K + K'" in exI)
         (use z y N False K in \<open>auto simp: add.assoc\<close>)
   qed
 qed
 
 lemmas multp_implies_one_step =
   mult_implies_one_step[of "{(x, y). r x y}" for r, folded multp_def transp_trans, simplified]
 
 lemma one_step_implies_mult:
   assumes
     "J \<noteq> {#}" and
     "\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> r"
   shows "(I + K, I + J) \<in> mult r"
   using assms
 proof (induction "size J" arbitrary: I J K)
   case 0
   then show ?case by auto
 next
   case (Suc n) note IH = this(1) and size_J = this(2)[THEN sym]
   obtain J' a where J: "J = add_mset a J'"
     using size_J by (blast dest: size_eq_Suc_imp_eq_union)
   show ?case
   proof (cases "J' = {#}")
     case True
     then show ?thesis
       using J Suc by (fastforce simp add: mult_def mult1_def)
   next
     case [simp]: False
     have K: "K = {#x \<in># K. (x, a) \<in> r#} + {#x \<in># K. (x, a) \<notin> r#}"
       by simp
     have "(I + K, (I + {# x \<in># K. (x, a) \<in> r #}) + J') \<in> mult r"
       using IH[of J' "{# x \<in># K. (x, a) \<notin> r#}" "I + {# x \<in># K. (x, a) \<in> r#}"]
         J Suc.prems K size_J by (auto simp: ac_simps)
     moreover have "(I + {#x \<in># K. (x, a) \<in> r#} + J', I + J) \<in> mult r"
       by (fastforce simp: J mult1_def mult_def)
     ultimately show ?thesis
       unfolding mult_def by simp
   qed
 qed
 
 lemmas one_step_implies_multp =
   one_step_implies_mult[of _ _ "{(x, y). r x y}" for r, folded multp_def, simplified]
 
 lemma subset_implies_mult:
   assumes sub: "A \<subset># B"
   shows "(A, B) \<in> mult r"
 proof -
   have ApBmA: "A + (B - A) = B"
     using sub by simp
   have BmA: "B - A \<noteq> {#}"
     using sub by (simp add: Diff_eq_empty_iff_mset subset_mset.less_le_not_le)
   thus ?thesis
     by (rule one_step_implies_mult[of "B - A" "{#}" _ A, unfolded ApBmA, simplified])
 qed
 
 lemmas subset_implies_multp = subset_implies_mult[of _ _ "{(x, y). r x y}" for r, folded multp_def]
 
 
 subsubsection \<open>The multiset extension is cancellative for multiset union\<close>
 
 lemma mult_cancel:
   assumes "trans s" and "irrefl s"
   shows "(X + Z, Y + Z) \<in> mult s \<longleftrightarrow> (X, Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
 proof
   assume ?L thus ?R
   proof (induct Z)
     case (add z Z)
     obtain X' Y' Z' where *: "add_mset z X + Z = Z' + X'" "add_mset z Y + Z = Z' + Y'" "Y' \<noteq> {#}"
       "\<forall>x \<in> set_mset X'. \<exists>y \<in> set_mset Y'. (x, y) \<in> s"
       using mult_implies_one_step[OF \<open>trans s\<close> add(2)] by auto
     consider Z2 where "Z' = add_mset z Z2" | X2 Y2 where "X' = add_mset z X2" "Y' = add_mset z Y2"
       using *(1,2) by (metis add_mset_remove_trivial_If insert_iff set_mset_add_mset_insert union_iff)
     thus ?case
     proof (cases)
       case 1 thus ?thesis using * one_step_implies_mult[of Y' X' s Z2]
         by (auto simp: add.commute[of _ "{#_#}"] add.assoc intro: add(1))
     next
       case 2 then obtain y where "y \<in> set_mset Y2" "(z, y) \<in> s" using *(4) \<open>irrefl s\<close>
         by (auto simp: irrefl_def)
       moreover from this transD[OF \<open>trans s\<close> _ this(2)]
       have "x' \<in> set_mset X2 \<Longrightarrow> \<exists>y \<in> set_mset Y2. (x', y) \<in> s" for x'
         using 2 *(4)[rule_format, of x'] by auto
       ultimately show ?thesis using  * one_step_implies_mult[of Y2 X2 s Z'] 2
         by (force simp: add.commute[of "{#_#}"] add.assoc[symmetric] intro: add(1))
     qed
   qed auto
 next
   assume ?R then obtain I J K
     where "Y = I + J" "X = I + K" "J \<noteq> {#}" "\<forall>k \<in> set_mset K. \<exists>j \<in> set_mset J. (k, j) \<in> s"
     using mult_implies_one_step[OF \<open>trans s\<close>] by blast
   thus ?L using one_step_implies_mult[of J K s "I + Z"] by (auto simp: ac_simps)
 qed
 
 lemmas multp_cancel =
   mult_cancel[of "{(x, y). r x y}" for r,
     folded multp_def transp_trans irreflp_irrefl_eq, simplified]
 
 lemmas mult_cancel_add_mset =
   mult_cancel[of _ _ "{#_#}", unfolded union_mset_add_mset_right add.comm_neutral]
 
 lemmas multp_cancel_add_mset =
   mult_cancel_add_mset[of "{(x, y). r x y}" for r,
     folded multp_def transp_trans irreflp_irrefl_eq, simplified]
 
 lemma mult_cancel_max0:
   assumes "trans s" and "irrefl s"
   shows "(X, Y) \<in> mult s \<longleftrightarrow> (X - X \<inter># Y, Y - X \<inter># Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
 proof -
   have "X - X \<inter># Y + X \<inter># Y = X" "Y - X \<inter># Y + X \<inter># Y = Y" by (auto simp flip: count_inject)
   thus ?thesis using mult_cancel[OF assms, of "X - X \<inter># Y"  "X \<inter># Y" "Y - X \<inter># Y"] by auto
 qed
 
 lemmas mult_cancel_max = mult_cancel_max0[simplified]
 
 lemmas multp_cancel_max =
   mult_cancel_max[of "{(x, y). r x y}" for r,
     folded multp_def transp_trans irreflp_irrefl_eq, simplified]
 
 
 subsubsection \<open>Partial-order properties\<close>
 
 lemma mult1_lessE:
   assumes "(N, M) \<in> mult1 {(a, b). r a b}" and "asymp r"
   obtains a M0 K where "M = add_mset a M0" "N = M0 + K"
     "a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> r b a"
 proof -
   from assms obtain a M0 K where "M = add_mset a M0" "N = M0 + K" and
     *: "b \<in># K \<Longrightarrow> r b a" for b by (blast elim: mult1E)
   moreover from * [of a] have "a \<notin># K"
     using \<open>asymp r\<close> by (meson asymp.cases)
   ultimately show thesis by (auto intro: that)
 qed
 
 lemma trans_mult: "trans r \<Longrightarrow> trans (mult r)"
   by (simp add: mult_def)
 
 lemma transp_multp: "transp r \<Longrightarrow> transp (multp r)"
   unfolding multp_def transp_trans_eq
   by (fact trans_mult[of "{(x, y). r x y}" for r, folded transp_trans])
 
 lemma irrefl_mult:
   assumes "trans r" "irrefl r"
   shows "irrefl (mult r)"
 proof (intro irreflI notI)
   fix M
   assume "(M, M) \<in> mult r"
   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> r)"
     using mult_implies_one_step[OF \<open>trans r\<close>] by blast
   then have *: "K \<noteq> {#}" and **: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. (k, j) \<in> r" by auto
   have "finite (set_mset K)" by simp
   hence "set_mset K = {}"
     using **
   proof (induction rule: finite_induct)
     case empty
     thus ?case by simp
   next
     case (insert x F)
     have False
       using \<open>irrefl r\<close>[unfolded irrefl_def, rule_format]
       using \<open>trans r\<close>[THEN transD]
       by (metis equals0D insert.IH insert.prems insertE insertI1 insertI2)
     thus ?case ..
   qed
   with * show False by simp
 qed
 
 lemmas irreflp_multp =
   irrefl_mult[of "{(x, y). r x y}" for r,
     folded transp_trans_eq irreflp_irrefl_eq, simplified, folded multp_def]
 
 instantiation multiset :: (preorder) order begin
 
 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"
   where "M < N \<longleftrightarrow> multp (<) M N"
 
 definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"
   where "less_eq_multiset M N \<longleftrightarrow> M < N \<or> M = N"
 
 instance
 proof intro_classes
   fix M N :: "'a multiset"
   show "(M < N) = (M \<le> N \<and> \<not> N \<le> M)"
     unfolding less_eq_multiset_def less_multiset_def
     by (metis irreflp_def irreflp_less irreflp_multp transpE transp_less transp_multp)
 next
   fix M :: "'a multiset"
   show "M \<le> M"
     unfolding less_eq_multiset_def
     by simp
 next
   fix M1 M2 M3 :: "'a multiset"
   show "M1 \<le> M2 \<Longrightarrow> M2 \<le> M3 \<Longrightarrow> M1 \<le> M3"
     unfolding less_eq_multiset_def less_multiset_def
     using transp_multp[OF transp_less, THEN transpD]
     by blast
 next
   fix M N :: "'a multiset"
   show "M \<le> N \<Longrightarrow> N \<le> M \<Longrightarrow> M = N"
     unfolding less_eq_multiset_def less_multiset_def
     using transp_multp[OF transp_less, THEN transpD]
     using irreflp_multp[OF transp_less irreflp_less, unfolded irreflp_def, rule_format]
     by blast
 qed
 
 end
 
 lemma mset_le_irrefl [elim!]:
   fixes M :: "'a::preorder multiset"
   shows "M < M \<Longrightarrow> R"
   by simp
 
 lemma wfP_less_multiset[simp]:
   assumes wfP_less: "wfP ((<) :: ('a :: preorder) \<Rightarrow> 'a \<Rightarrow> bool)"
   shows "wfP ((<) :: 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool)"
   using wfP_multp[OF wfP_less] less_multiset_def
   by (metis wfPUNIVI wfP_induct)
 
 
 subsection \<open>Quasi-executable version of the multiset extension\<close>
 
 text \<open>
   Predicate variants of \<open>mult\<close> and the reflexive closure of \<open>mult\<close>, which are
   executable whenever the given predicate \<open>P\<close> is. Together with the standard
   code equations for \<open>(\<inter>#\<close>) and \<open>(-\<close>) this should yield quadratic
   (with respect to calls to \<open>P\<close>) implementations of \<open>multp_code\<close> and \<open>multeqp_code\<close>.
 \<close>
 
 definition multp_code :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multp_code P N M =
     (let Z = M \<inter># N; X = M - Z in
     X \<noteq> {#} \<and> (let Y = N - Z in (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x)))"
 
 definition multeqp_code :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
   "multeqp_code P N M =
     (let Z = M \<inter># N; X = M - Z; Y = N - Z in
     (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x))"
 
 lemma multp_code_iff_mult:
   assumes "irrefl R" and "trans R" and [simp]: "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
   shows "multp_code P N M \<longleftrightarrow> (N, M) \<in> mult R" (is "?L \<longleftrightarrow> ?R")
 proof -
   have *: "M \<inter># N + (N - M \<inter># N) = N" "M \<inter># N + (M - M \<inter># N) = M"
     "(M - M \<inter># N) \<inter># (N - M \<inter># N) = {#}" by (auto simp flip: count_inject)
   show ?thesis
   proof
     assume ?L thus ?R
       using one_step_implies_mult[of "M - M \<inter># N" "N - M \<inter># N" R "M \<inter># N"] *
       by (auto simp: multp_code_def Let_def)
   next
     { fix I J K :: "'a multiset" assume "(I + J) \<inter># (I + K) = {#}"
       then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty)
     } note [dest!] = this
     assume ?R thus ?L
       using mult_implies_one_step[OF assms(2), of "N - M \<inter># N" "M - M \<inter># N"]
         mult_cancel_max[OF assms(2,1), of "N" "M"] * by (auto simp: multp_code_def)
   qed
 qed
 
 lemma multp_code_eq_multp: "irreflp r \<Longrightarrow> transp r \<Longrightarrow> multp_code r = multp r"
   using multp_code_iff_mult[of "{(x, y). r x y}" r for r,
     folded irreflp_irrefl_eq transp_trans multp_def, simplified]
   by blast
 
 lemma multeqp_code_iff_reflcl_mult:
   assumes "irrefl R" and "trans R" and "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
   shows "multeqp_code P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>="
 proof -
   { assume "N \<noteq> M" "M - M \<inter># N = {#}"
     then obtain y where "count N y \<noteq> count M y" by (auto simp flip: count_inject)
     then have "\<exists>y. count M y < count N y" using \<open>M - M \<inter># N = {#}\<close>
       by (auto simp flip: count_inject dest!: le_neq_implies_less fun_cong[of _ _ y])
   }
   then have "multeqp_code P N M \<longleftrightarrow> multp_code P N M \<or> N = M"
     by (auto simp: multeqp_code_def multp_code_def Let_def in_diff_count)
   thus ?thesis using multp_code_iff_mult[OF assms] by simp
 qed
 
 lemma multeqp_code_eq_reflclp_multp: "irreflp r \<Longrightarrow> transp r \<Longrightarrow> multeqp_code r = (multp r)\<^sup>=\<^sup>="
   using multeqp_code_iff_reflcl_mult[of "{(x, y). r x y}" r for r,
     folded irreflp_irrefl_eq transp_trans, simplified, folded multp_def]
   by blast
 
 
 subsubsection \<open>Monotonicity of multiset union\<close>
 
 lemma mult1_union: "(B, D) \<in> mult1 r \<Longrightarrow> (C + B, C + D) \<in> mult1 r"
   by (force simp: mult1_def)
 
 lemma union_le_mono2: "B < D \<Longrightarrow> C + B < C + (D::'a::preorder multiset)"
 apply (unfold less_multiset_def multp_def mult_def)
 apply (erule trancl_induct)
  apply (blast intro: mult1_union)
 apply (blast intro: mult1_union trancl_trans)
 done
 
 lemma union_le_mono1: "B < D \<Longrightarrow> B + C < D + (C::'a::preorder multiset)"
 apply (subst add.commute [of B C])
 apply (subst add.commute [of D C])
 apply (erule union_le_mono2)
 done
 
 lemma union_less_mono:
   fixes A B C D :: "'a::preorder multiset"
   shows "A < C \<Longrightarrow> B < D \<Longrightarrow> A + B < C + D"
   by (blast intro!: union_le_mono1 union_le_mono2 less_trans)
 
 instantiation multiset :: (preorder) ordered_ab_semigroup_add
 begin
 instance
   by standard (auto simp add: less_eq_multiset_def intro: union_le_mono2)
 end
 
 
 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
     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
     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>\<open>multiset T\<close>;
 
     fun mk_mset T [] = \<^instantiate>\<open>'a = T in term \<open>{#}\<close>\<close>
       | mk_mset T [x] = \<^instantiate>\<open>'a = T and x in term \<open>{#x#}\<close>\<close>
       | mk_mset T (x :: xs) = \<^Const>\<open>plus \<open>msetT T\<close> for \<open>mk_mset T [x]\<close> \<open>mk_mset T xs\<close>\<close>
 
     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>\<open>empty_mset\<close> \<^const_name>\<open>plus\<close>
         (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 add_mset_commute
 
 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_subset_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 mset_le_not_refl: "\<not> M < (M::'a::preorder multiset)"
   by (fact less_irrefl)
 
 lemma mset_le_trans: "K < M \<Longrightarrow> M < N \<Longrightarrow> K < (N::'a::preorder multiset)"
   by (fact less_trans)
 
 lemma mset_le_not_sym: "M < N \<Longrightarrow> \<not> N < (M::'a::preorder multiset)"
   by (fact less_not_sym)
 
 lemma mset_le_asym: "M < N \<Longrightarrow> (\<not> P \<Longrightarrow> N < (M::'a::preorder multiset)) \<Longrightarrow> P"
   by (fact 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'
               ||> (~~)
               ||> map (apsnd (snd o HOLogic.dest_number))
             fun elems_for t =
               (case AList.lookup (=) 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>\<open>Groups.zero T\<close>
             | ts => foldl1 (fn (s, t) => \<^Const>\<open>add_mset elem_T for s t\<close>) ts)
           end
       | multiset_postproc _ _ _ _ t = t
   in Nitpick_Model.register_term_postprocessor \<^typ>\<open>'a multiset\<close> multiset_postproc end
 \<close>
 
 
 subsection \<open>Naive implementation using lists\<close>
 
 code_datatype mset
 
 lemma [code]: "{#} = mset []"
   by simp
 
 lemma [code]: "add_mset x (mset xs) = mset (x # xs)"
   by simp
 
 lemma [code]: "Multiset.is_empty (mset xs) \<longleftrightarrow> List.null xs"
   by (simp add: Multiset.is_empty_def List.null_def)
 
 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
 
 lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"
   by simp
 
 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 diff_diff_add)
 
 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
   done
 
 declare size_mset [code]
 
 fun subset_eq_mset_impl :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool option" where
   "subset_eq_mset_impl [] ys = Some (ys \<noteq> [])"
 | "subset_eq_mset_impl (Cons x xs) ys = (case List.extract ((=) x) ys of
      None \<Rightarrow> None
    | Some (ys1,_,ys2) \<Rightarrow> subset_eq_mset_impl xs (ys1 @ ys2))"
 
 lemma subset_eq_mset_impl: "(subset_eq_mset_impl xs ys = None \<longleftrightarrow> \<not> mset xs \<subseteq># mset ys) \<and>
   (subset_eq_mset_impl xs ys = Some True \<longleftrightarrow> mset xs \<subset># mset ys) \<and>
   (subset_eq_mset_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
 proof (induct xs arbitrary: ys)
   case (Nil ys)
   show ?case by (auto simp: subset_mset.zero_less_iff_neq_zero)
 next
   case (Cons x xs ys)
   show ?case
   proof (cases "List.extract ((=) 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 = add_mset x (mset (ys1 @ ys2))"
       by auto
     show ?thesis unfolding subset_eq_mset_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> subset_eq_mset_impl xs ys \<noteq> None"
   using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)
 
 lemma [code]: "mset xs \<subset># mset ys \<longleftrightarrow> subset_eq_mset_impl xs ys = Some True"
   using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_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> subset_eq_mset_impl xs ys = Some False"
   unfolding equal_multiset_def
   using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)
 
 instance
   by standard (simp add: equal_multiset_def)
 
 end
 
 declare sum_mset_sum_list [code]
 
 lemma [code]: "prod_mset (mset xs) = fold times xs 1"
 proof -
   have "\<And>x. fold times xs x = prod_mset (mset xs) * x"
     by (induct xs) (simp_all add: ac_simps)
   then show ?thesis by simp
 qed
 
 text \<open>
   Exercise for the casual reader: add implementations for \<^term>\<open>(\<le>)\<close>
   and \<^term>\<open>(<)\<close> (multiset order).
 \<close>
 
 text \<open>Quickcheck generators\<close>
 
 context
   includes term_syntax
 begin
 
 definition
   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"
 
 end
 
 instantiation multiset :: (random) random
 begin
 
 context
   includes state_combinator_syntax
 begin
 
 definition
   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (msetify xs))"
 
 instance ..
 
 end
 
 end
 
 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)) =
     add_mset (x,y) (mset (zip xs ys))"
   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)) =
       add_mset (x,y) (mset (zip xs ys))"
       by (rule Cons.hyps(2))
     thus ?thesis
       unfolding k by auto
   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)
 
   define xsa where "xsa = take j xs' @ drop (Suc j) xs'"
   have "mset xs' = {#x#} + mset xsa"
     unfolding xsa_def using j_len nth_j
     by (metis Cons_nth_drop_Suc union_mset_add_mset_right add_mset_remove_trivial add_diff_cancel_left'
         append_take_drop_id mset.simps(2) mset_append)
   hence ms_x: "mset xsa = mset xs"
     by (simp add: Cons.prems)
   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
 
   define ys' where "ys' = 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')
   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 (add_mset a M)"
 
 lemma pred_mset_iff: \<comment> \<open>TODO: alias for \<^const>\<open>Multiset.Ball\<close>\<close>
   \<open>pred_mset P M \<longleftrightarrow> Multiset.Ball M P\<close>  (is \<open>?P \<longleftrightarrow> ?Q\<close>)
 proof
   assume ?P
   then show ?Q by induction simp_all
 next
   assume ?Q
   then show ?P
     by (induction M) (auto intro: pred_mset.intros)
 qed
 
 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 = (`) 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])
   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 simp flip: mset_map)
     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
     by (simp add: fun_eq_iff pred_mset_iff)
 qed
 
 inductive rel_mset' :: \<open>('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset \<Rightarrow> bool\<close>
 where
   Zero[intro]: "rel_mset' R {#} {#}"
 | Plus[intro]: "\<lbrakk>R a b; rel_mset' R M N\<rbrakk> \<Longrightarrow> rel_mset' R (add_mset a M) (add_mset b N)"
 
 lemma rel_mset_Zero: "rel_mset R {#} {#}"
 unfolding rel_mset_def Grp_def by auto
 
 declare multiset.count[simp]
 declare count_Abs_multiset[simp]
 declare multiset.count_inverse[simp]
 
 lemma rel_mset_Plus:
   assumes ab: "R a b"
     and MN: "rel_mset R M N"
   shows "rel_mset R (add_mset a M) (add_mset b N)"
 proof -
   have "\<exists>ya. add_mset a (image_mset fst y) = image_mset fst ya \<and>
     add_mset b (image_mset snd y) = 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 _ "add_mset (a,b) y"]) 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 rel_mset_Zero_iff [simp]:
   shows "rel_mset rel {#} Y \<longleftrightarrow> Y = {#}" and "rel_mset rel X {#} \<longleftrightarrow> X = {#}"
   by (auto simp add: rel_mset_Zero dest: rel_mset_size)
 
 lemma multiset_induct2[case_names empty addL addR]:
   assumes empty: "P {#} {#}"
     and addL: "\<And>a M N. P M N \<Longrightarrow> P (add_mset a M) N"
     and addR: "\<And>a M N. P M N \<Longrightarrow> P M (add_mset a N)"
   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>a b M N a b. P M N \<Longrightarrow> P (add_mset a M) (add_mset b N)"
   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 = add_mset a M1" by (metis multi_nonempty_split)
     have "N \<noteq> {#}" using False less.prems by auto
     then obtain N1 b where N: "N = add_mset b N1" 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 (add_mset a M) = N"
   shows "\<exists>N1. N = add_mset (f a) N1 \<and> image_mset f M = N1"
 proof -
   have "f a \<in># N"
     using assms multiset.set_map[of f "add_mset a M"] by auto
   then obtain N1 where N: "N = add_mset (f a) N1" 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 = add_mset b N"
   shows "\<exists>M1 a. M = add_mset a M1 \<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 = add_mset a M1" 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 (add_mset a M) N"
   shows "\<exists>N1 b. N = add_mset b N1 \<and> R a b \<and> rel_mset R M N1"
 proof -
   obtain K where KM: "image_mset fst K = add_mset a M"
     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 = add_mset ab K1" 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 = add_mset (snd ab) N1" 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 (add_mset b N)"
   shows "\<exists>M1 a. M = add_mset a M1 \<and> R a b \<and> rel_mset R M1 N"
 proof -
   obtain K where KN: "image_mset snd K = add_mset b N"
     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 = add_mset ab K1" 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 = add_mset (fst ab) M1" 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 = add_mset a M1" by (metis multi_nonempty_split)
     obtain N1 b where N: "N = add_mset b N1" 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>\<open>rel_mset\<close>: 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 size_multiset_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>\<open>multiset\<close> \<^const_name>\<open>size_multiset\<close>
     @{thm size_multiset_overloaded_def}
     @{thms size_multiset_empty size_multiset_single size_multiset_union size_empty size_single
       size_union}
     @{thms size_multiset_o_map}
 \<close>
 
 subsection \<open>Lemmas about Size\<close>
 
 lemma size_mset_SucE: "size A = Suc n \<Longrightarrow> (\<And>a B. A = {#a#} + B \<Longrightarrow> size B = n \<Longrightarrow> P) \<Longrightarrow> P"
   by (cases A) (auto simp add: ac_simps)
 
 lemma size_Suc_Diff1: "x \<in># M \<Longrightarrow> Suc (size (M - {#x#})) = size M"
   using arg_cong[OF insert_DiffM, of _ _ size] by simp
 
 lemma size_Diff_singleton: "x \<in># M \<Longrightarrow> size (M - {#x#}) = size M - 1"
   by (simp flip: size_Suc_Diff1)
 
 lemma size_Diff_singleton_if: "size (A - {#x#}) = (if x \<in># A then size A - 1 else size A)"
   by (simp add: diff_single_trivial size_Diff_singleton)
 
 lemma size_Un_Int: "size A + size B = size (A \<union># B) + size (A \<inter># B)"
   by (metis inter_subset_eq_union size_union subset_mset.diff_add union_diff_inter_eq_sup)
 
 lemma size_Un_disjoint: "A \<inter># B = {#} \<Longrightarrow> size (A \<union># B) = size A + size B"
   using size_Un_Int[of A B] by simp
 
 lemma size_Diff_subset_Int: "size (M - M') = size M - size (M \<inter># M')"
   by (metis diff_intersect_left_idem size_Diff_submset subset_mset.inf_le1)
 
 lemma diff_size_le_size_Diff: "size (M :: _ multiset) - size M' \<le> size (M - M')"
   by (simp add: diff_le_mono2 size_Diff_subset_Int size_mset_mono)
 
 lemma size_Diff1_less: "x\<in># M \<Longrightarrow> size (M - {#x#}) < size M"
   by (rule Suc_less_SucD) (simp add: size_Suc_Diff1)
 
 lemma size_Diff2_less: "x\<in># M \<Longrightarrow> y\<in># M \<Longrightarrow> size (M - {#x#} - {#y#}) < size M"
   by (metis less_imp_diff_less size_Diff1_less size_Diff_subset_Int)
 
 lemma size_Diff1_le: "size (M - {#x#}) \<le> size M"
   by (cases "x \<in># M") (simp_all add: size_Diff1_less less_imp_le diff_single_trivial)
 
 lemma size_psubset: "M \<subseteq># M' \<Longrightarrow> size M < size M' \<Longrightarrow> M \<subset># M'"
   using less_irrefl subset_mset_def by blast
 
 hide_const (open) wcount
 
 end