(*  Title:      HOL/Library/Multiset.thy

     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker

 *)

 header {* (Finite) multisets *}

 theory Multiset

 imports Main

 begin

 subsection {* The type of multisets *}

 typedef 'a multiset = "{f :: 'a => nat. finite {x. f x > 0}}"

   morphisms count Abs_multiset

 proof

   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp

 qed

 lemmas multiset_typedef = Abs_multiset_inverse count_inverse count

 abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where

   "a :# M == 0 < count M a"

 notation (xsymbols)

   Melem (infix "\<in>#" 50)

 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 {*

  \medskip Preservation of the representing set @{term multiset}.

 *}

 lemma const0_in_multiset:

   "(\<lambda>a. 0) \<in> multiset"

   by (simp add: multiset_def)

 lemma only1_in_multiset:

   "(\<lambda>b. if b = a then n else 0) \<in> multiset"

   by (simp add: multiset_def)

 lemma union_preserves_multiset:

   "M \<in> multiset \<Longrightarrow> N \<in> multiset \<Longrightarrow> (\<lambda>a. M a + N a) \<in> multiset"

   by (simp add: multiset_def)

 lemma diff_preserves_multiset:

   assumes "M \<in> multiset"

   shows "(\<lambda>a. M a - N a) \<in> multiset"

 proof -

   have "{x. N x < M x} \<subseteq> {x. 0 < M x}"

     by auto

   with assms show ?thesis

     by (auto simp add: multiset_def intro: finite_subset)

 qed

 lemma filter_preserves_multiset:

   assumes "M \<in> multiset"

   shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"

 proof -

   have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"

     by auto

   with assms show ?thesis

     by (auto simp add: multiset_def intro: finite_subset)

 qed

 lemmas in_multiset = const0_in_multiset only1_in_multiset

   union_preserves_multiset diff_preserves_multiset filter_preserves_multiset

 subsection {* Representing multisets *}

 text {* Multiset enumeration *}

 instantiation multiset :: (type) "{zero, plus}"

 begin

 definition Mempty_def:

   "0 = Abs_multiset (\<lambda>a. 0)"

 abbreviation Mempty :: "'a multiset" ("{#}") where

   "Mempty \<equiv> 0"

 definition union_def:

   "M + N = Abs_multiset (\<lambda>a. count M a + count N a)"

 instance ..

 end

 definition single :: "'a => 'a multiset" where

   "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"

 syntax

   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")

 translations

   "{#x, xs#}" == "{#x#} + {#xs#}"

   "{#x#}" == "CONST single x"

 lemma count_empty [simp]: "count {#} a = 0"

   by (simp add: Mempty_def in_multiset multiset_typedef)

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

   by (simp add: single_def in_multiset multiset_typedef)

 subsection {* Basic operations *}

 subsubsection {* Union *}

 lemma count_union [simp]: "count (M + N) a = count M a + count N a"

   by (simp add: union_def in_multiset multiset_typedef)

 instance multiset :: (type) cancel_comm_monoid_add proof

 qed (simp_all add: multiset_eq_iff)

 subsubsection {* Difference *}

 instantiation multiset :: (type) minus

 begin

 definition diff_def:

   "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"

 instance ..

 end

 lemma count_diff [simp]: "count (M - N) a = count M a - count N a"

   by (simp add: diff_def in_multiset multiset_typedef)

 lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"

 by(simp add: multiset_eq_iff)

 lemma diff_cancel[simp]: "A - A = {#}"

 by (rule multiset_eqI) simp

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

 by(simp add: multiset_eq_iff)

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

 by(simp add: multiset_eq_iff)

 lemma insert_DiffM:

   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"

   by (clarsimp simp: multiset_eq_iff)

 lemma insert_DiffM2 [simp]:

   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"

   by (clarsimp simp: multiset_eq_iff)

 lemma diff_right_commute:

   "(M::'a multiset) - N - Q = M - Q - N"

   by (auto simp add: multiset_eq_iff)

 lemma diff_add:

   "(M::'a multiset) - (N + Q) = M - N - Q"

 by (simp add: multiset_eq_iff)

 lemma diff_union_swap:

   "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"

   by (auto simp add: multiset_eq_iff)

 lemma diff_union_single_conv:

   "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"

   by (simp add: multiset_eq_iff)

 subsubsection {* Equality of multisets *}

 lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"

   by (simp add: multiset_eq_iff)

 lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"

   by (auto simp add: multiset_eq_iff)

 lemma union_eq_empty [iff]: "M + N = {#} \<longleftrightarrow> M = {#} \<and> N = {#}"

   by (auto simp add: multiset_eq_iff)

 lemma empty_eq_union [iff]: "{#} = M + N \<longleftrightarrow> M = {#} \<and> N = {#}"

   by (auto simp add: multiset_eq_iff)

 lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"

   by (auto simp add: multiset_eq_iff)

 lemma diff_single_trivial:

   "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"

   by (auto simp add: multiset_eq_iff)

 lemma diff_single_eq_union:

   "x \<in># M \<Longrightarrow> M - {#x#} = N \<longleftrightarrow> M = N + {#x#}"

   by auto

 lemma union_single_eq_diff:

   "M + {#x#} = N \<Longrightarrow> M = N - {#x#}"

   by (auto dest: sym)

 lemma union_single_eq_member:

   "M + {#x#} = N \<Longrightarrow> x \<in># N"

   by auto

 lemma union_is_single:

   "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")proof

   assume ?rhs then show ?lhs by auto

 next

   assume ?lhs thus ?rhs

     by(simp add: multiset_eq_iff split:if_splits) (metis add_is_1)

 qed

 lemma single_is_union:

   "{#a#} = M + N \<longleftrightarrow> {#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N"

   by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)

 lemma add_eq_conv_diff:

   "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")

 (* shorter: by (simp add: multiset_eq_iff) fastsimp *)

 proof

   assume ?rhs then show ?lhs

   by (auto simp add: add_assoc add_commute [of "{#b#}"])

     (drule sym, simp add: add_assoc [symmetric])

 next

   assume ?lhs

   show ?rhs

   proof (cases "a = b")

     case True with `?lhs` show ?thesis by simp

   next

     case False

     from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)

     with False have "a \<in># N" by auto

     moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)

     moreover note False

     ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)

   qed

 qed

 lemma insert_noteq_member: 

   assumes BC: "B + {#b#} = C + {#c#}"

    and bnotc: "b \<noteq> c"

   shows "c \<in># B"

 proof -

   have "c \<in># C + {#c#}" by simp

   have nc: "\<not> c \<in># {#b#}" using bnotc by simp

   then have "c \<in># B + {#b#}" using BC by simp

   then show "c \<in># B" using nc by simp

 qed

 lemma add_eq_conv_ex:

   "(M + {#a#} = N + {#b#}) =

     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"

   by (auto simp add: add_eq_conv_diff)

 subsubsection {* Pointwise ordering induced by count *}

 instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le

 begin

 definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where

   mset_le_def: "A \<le> B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"

 definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where

   mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"

 instance proof

 qed (auto simp add: mset_le_def mset_less_def multiset_eq_iff intro: order_trans antisym)

 end

 lemma mset_less_eqI:

   "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"

   by (simp add: mset_le_def)

 lemma mset_le_exists_conv:

   "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"

 apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)

 apply (auto intro: multiset_eq_iff [THEN iffD2])

 done

 lemma mset_le_mono_add_right_cancel [simp]:

   "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"

   by (fact add_le_cancel_right)

 lemma mset_le_mono_add_left_cancel [simp]:

   "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"

   by (fact add_le_cancel_left)

 lemma mset_le_mono_add:

   "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"

   by (fact add_mono)

 lemma mset_le_add_left [simp]:

   "(A::'a multiset) \<le> A + B"

   unfolding mset_le_def by auto

 lemma mset_le_add_right [simp]:

   "B \<le> (A::'a multiset) + B"

   unfolding mset_le_def by auto

 lemma mset_le_single:

   "a :# B \<Longrightarrow> {#a#} \<le> B"

   by (simp add: mset_le_def)

 lemma multiset_diff_union_assoc:

   "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"

   by (simp add: multiset_eq_iff mset_le_def)

 lemma mset_le_multiset_union_diff_commute:

   "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"

 by (simp add: multiset_eq_iff mset_le_def)

 lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"

 by(simp add: mset_le_def)

 lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"

 apply (clarsimp simp: mset_le_def mset_less_def)

 apply (erule_tac x=x in allE)

 apply auto

 done

 lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"

 apply (clarsimp simp: mset_le_def mset_less_def)

 apply (erule_tac x = x in allE)

 apply auto

 done

 lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"

 apply (rule conjI)

  apply (simp add: mset_lessD)

 apply (clarsimp simp: mset_le_def mset_less_def)

 apply safe

  apply (erule_tac x = a in allE)

  apply (auto split: split_if_asm)

 done

 lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"

 apply (rule conjI)

  apply (simp add: mset_leD)

 apply (force simp: mset_le_def mset_less_def split: split_if_asm)

 done

 lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"

   by (auto simp add: mset_less_def mset_le_def multiset_eq_iff)

 lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"

   by (auto simp: mset_le_def mset_less_def)

 lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"

   by simp

 lemma mset_less_add_bothsides:

   "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"

   by (fact add_less_imp_less_right)

 lemma mset_less_empty_nonempty:

   "{#} < S \<longleftrightarrow> S \<noteq> {#}"

   by (auto simp: mset_le_def mset_less_def)

 lemma mset_less_diff_self:

   "c \<in># B \<Longrightarrow> B - {#c#} < B"

   by (auto simp: mset_le_def mset_less_def multiset_eq_iff)

 subsubsection {* Intersection *}

 instantiation multiset :: (type) semilattice_inf

 begin

 definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where

   multiset_inter_def: "inf_multiset A B = A - (A - B)"

 instance proof -

   have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith

   show "OFCLASS('a multiset, semilattice_inf_class)" proof

   qed (auto simp add: multiset_inter_def mset_le_def aux)

 qed

 end

 abbreviation multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where

   "multiset_inter \<equiv> inf"

 lemma multiset_inter_count [simp]:

   "count (A #\<inter> B) x = min (count A x) (count B x)"

   by (simp add: multiset_inter_def multiset_typedef)

 lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"

   by (rule multiset_eqI) (auto simp add: multiset_inter_count)

 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_inter_count multiset_eq_iff)

   then have "count B x = 0 \<or> count C x = 0"

     by auto

   then show "count (A + B - C) x = count (A - C + B) x"

     by auto

 qed

 subsubsection {* Filter (with comprehension syntax) *}

 text {* Multiset comprehension *}

 definition filter :: "('a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where

   "filter P M = Abs_multiset (\<lambda>x. if P x then count M x else 0)"

 hide_const (open) filter

 lemma count_filter [simp]:

   "count (Multiset.filter P M) a = (if P a then count M a else 0)"

   by (simp add: filter_def in_multiset multiset_typedef)

 lemma filter_empty [simp]:

   "Multiset.filter P {#} = {#}"

   by (rule multiset_eqI) simp

 lemma filter_single [simp]:

   "Multiset.filter P {#x#} = (if P x then {#x#} else {#})"

   by (rule multiset_eqI) simp

 lemma filter_union [simp]:

   "Multiset.filter P (M + N) = Multiset.filter P M + Multiset.filter P N"

   by (rule multiset_eqI) simp

 lemma filter_diff [simp]:

   "Multiset.filter P (M - N) = Multiset.filter P M - Multiset.filter P N"

   by (rule multiset_eqI) simp

 lemma filter_inter [simp]:

   "Multiset.filter P (M #\<inter> N) = Multiset.filter P M #\<inter> Multiset.filter P N"

   by (rule multiset_eqI) simp

 syntax

   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ :# _./ _#})")

 syntax (xsymbol)

   "_MCollect" :: "pttrn \<Rightarrow> 'a multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"    ("(1{# _ \<in># _./ _#})")

 translations

   "{#x \<in># M. P#}" == "CONST Multiset.filter (\<lambda>x. P) M"

 subsubsection {* Set of elements *}

 definition set_of :: "'a multiset => 'a set" where

   "set_of M = {x. x :# M}"

 lemma set_of_empty [simp]: "set_of {#} = {}"

 by (simp add: set_of_def)

 lemma set_of_single [simp]: "set_of {#b#} = {b}"

 by (simp add: set_of_def)

 lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"

 by (auto simp add: set_of_def)

 lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"

 by (auto simp add: set_of_def multiset_eq_iff)

 lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"

 by (auto simp add: set_of_def)

 lemma set_of_filter [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"

 by (auto simp add: set_of_def)

 lemma finite_set_of [iff]: "finite (set_of M)"

   using count [of M] by (simp add: multiset_def set_of_def)

 subsubsection {* Size *}

 instantiation multiset :: (type) size

 begin

 definition size_def:

   "size M = setsum (count M) (set_of M)"

 instance ..

 end

 lemma size_empty [simp]: "size {#} = 0"

 by (simp add: size_def)

 lemma size_single [simp]: "size {#b#} = 1"

 by (simp add: size_def)

 lemma setsum_count_Int:

   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"

 apply (induct rule: finite_induct)

  apply simp

 apply (simp add: Int_insert_left set_of_def)

 done

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

 apply (unfold size_def)

 apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")

  prefer 2

  apply (rule ext, simp)

 apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)

 apply (subst Int_commute)

 apply (simp (no_asm_simp) add: setsum_count_Int)

 done

 lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"

 by (auto simp add: size_def multiset_eq_iff)

 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 ==> \<exists>a. a :# M"

 apply (unfold size_def)

 apply (drule setsum_SucD)

 apply auto

 done

 lemma size_eq_Suc_imp_eq_union:

   assumes "size M = Suc n"

   shows "\<exists>a N. M = N + {#a#}"

 proof -

   from assms obtain a where "a \<in># M"

     by (erule size_eq_Suc_imp_elem [THEN exE])

   then have "M = M - {#a#} + {#a#}" by simp

   then show ?thesis by blast

 qed

 subsection {* Induction and case splits *}

 lemma setsum_decr:

   "finite F ==> (0::nat) < f a ==>

     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"

 apply (induct rule: finite_induct)

  apply auto

 apply (drule_tac a = a in mk_disjoint_insert, auto)

 done

 lemma rep_multiset_induct_aux:

 assumes 1: "P (\<lambda>a. (0::nat))"

   and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"

 shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"

 apply (unfold multiset_def)

 apply (induct_tac n, simp, clarify)

  apply (subgoal_tac "f = (\<lambda>a.0)")

   apply simp

   apply (rule 1)

  apply (rule ext, force, clarify)

 apply (frule setsum_SucD, clarify)

 apply (rename_tac a)

 apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")

  prefer 2

  apply (rule finite_subset)

   prefer 2

   apply assumption

  apply simp

  apply blast

 apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")

  prefer 2

  apply (rule ext)

  apply (simp (no_asm_simp))

  apply (erule ssubst, rule 2 [unfolded multiset_def], blast)

 apply (erule allE, erule impE, erule_tac [2] mp, blast)

 apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)

 apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")

  prefer 2

  apply blast

 apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")

  prefer 2

  apply blast

 apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)

 done

 theorem rep_multiset_induct:

   "f \<in> multiset ==> P (\<lambda>a. 0) ==>

     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"

 using rep_multiset_induct_aux by blast

 theorem multiset_induct [case_names empty add, induct type: multiset]:

 assumes empty: "P {#}"

   and add: "!!M x. P M ==> P (M + {#x#})"

 shows "P M"

 proof -

   note defns = union_def single_def Mempty_def

   note add' = add [unfolded defns, simplified]

   have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) =

     (\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset) 

   show ?thesis

     apply (rule count_inverse [THEN subst])

     apply (rule count [THEN rep_multiset_induct])

      apply (rule empty [unfolded defns])

     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")

      prefer 2

      apply (simp add: fun_eq_iff)

     apply (erule ssubst)

     apply (erule Abs_multiset_inverse [THEN subst])

     apply (drule add')

     apply (simp add: aux)

     done

 qed

 lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"

 by (induct M) auto

 lemma multiset_cases [cases type, case_names empty add]:

 assumes em:  "M = {#} \<Longrightarrow> P"

 assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"

 shows "P"

 proof (cases "M = {#}")

   assume "M = {#}" then show ?thesis using em by simp

 next

   assume "M \<noteq> {#}"

   then obtain M' m where "M = M' + {#m#}" 

     by (blast dest: multi_nonempty_split)

   then show ?thesis using add by simp

 qed

 lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"

 apply (cases M)

  apply simp

 apply (rule_tac x="M - {#x#}" in exI, simp)

 done

 lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"

 by (cases "B = {#}") (auto dest: multi_member_split)

 lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"

 apply (subst multiset_eq_iff)

 apply auto

 done

 lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"

 proof (induct A arbitrary: B)

   case (empty M)

   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)

   then obtain M' x where "M = M' + {#x#}" 

     by (blast dest: multi_nonempty_split)

   then show ?case by simp

 next

   case (add S x T)

   have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact

   have SxsubT: "S + {#x#} < T" by fact

   then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)

   then obtain T' where T: "T = T' + {#x#}" 

     by (blast dest: multi_member_split)

   then have "S < T'" using SxsubT 

     by (blast intro: mset_less_add_bothsides)

   then have "size S < size T'" using IH by simp

   then show ?case using T by simp

 qed

 subsubsection {* Strong induction and subset induction for multisets *}

 text {* Well-foundedness of proper subset operator: *}

 text {* proper multiset subset *}

 definition

   mset_less_rel :: "('a multiset * 'a multiset) set" where

   "mset_less_rel = {(A,B). A < B}"

 lemma multiset_add_sub_el_shuffle: 

   assumes "c \<in># B" and "b \<noteq> c" 

   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"

 proof -

   from `c \<in># B` obtain A where B: "B = A + {#c#}" 

     by (blast dest: multi_member_split)

   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp

   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 

     by (simp add: add_ac)

   then show ?thesis using B by simp

 qed

 lemma wf_mset_less_rel: "wf mset_less_rel"

 apply (unfold mset_less_rel_def)

 apply (rule wf_measure [THEN wf_subset, where f1=size])

 apply (clarsimp simp: measure_def inv_image_def mset_less_size)

 done

 text {* The induction rules: *}

 lemma full_multiset_induct [case_names less]:

 assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"

 shows "P B"

 apply (rule wf_mset_less_rel [THEN wf_induct])

 apply (rule ih, auto simp: mset_less_rel_def)

 done

 lemma multi_subset_induct [consumes 2, case_names empty add]:

 assumes "F \<le> A"

   and empty: "P {#}"

   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"

 shows "P F"

 proof -

   from `F \<le> A`

   show ?thesis

   proof (induct F)

     show "P {#}" by fact

   next

     fix x F

     assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"

     show "P (F + {#x#})"

     proof (rule insert)

       from i show "x \<in># A" by (auto dest: mset_le_insertD)

       from i have "F \<le> A" by (auto dest: mset_le_insertD)

       with P show "P F" .

     qed

   qed

 qed

 subsection {* Alternative representations *}

 subsubsection {* Lists *}

 primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where

   "multiset_of [] = {#}" |

   "multiset_of (a # x) = multiset_of x + {# a #}"

 lemma in_multiset_in_set:

   "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"

   by (induct xs) simp_all

 lemma count_multiset_of:

   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"

   by (induct xs) simp_all

 lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"

 by (induct x) auto

 lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"

 by (induct x) auto

 lemma set_of_multiset_of[simp]: "set_of (multiset_of x) = set x"

 by (induct x) auto

 lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"

 by (induct xs) auto

 lemma multiset_of_append [simp]:

   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"

   by (induct xs arbitrary: ys) (auto simp: add_ac)

 lemma multiset_of_filter:

   "multiset_of (filter P xs) = {#x :# multiset_of xs. P x #}"

   by (induct xs) simp_all

 lemma multiset_of_rev [simp]:

   "multiset_of (rev xs) = multiset_of xs"

   by (induct xs) simp_all

 lemma surj_multiset_of: "surj multiset_of"

 apply (unfold surj_def)

 apply (rule allI)

 apply (rule_tac M = y in multiset_induct)

  apply auto

 apply (rule_tac x = "x # xa" in exI)

 apply auto

 done

 lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"

 by (induct x) auto

 lemma distinct_count_atmost_1:

   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"

 apply (induct x, simp, rule iffI, simp_all)

 apply (rule conjI)

 apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)

 apply (erule_tac x = a in allE, simp, clarify)

 apply (erule_tac x = aa in allE, simp)

 done

 lemma multiset_of_eq_setD:

   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"

 by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)

 lemma set_eq_iff_multiset_of_eq_distinct:

   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>

     (set x = set y) = (multiset_of x = multiset_of y)"

 by (auto simp: multiset_eq_iff distinct_count_atmost_1)

 lemma set_eq_iff_multiset_of_remdups_eq:

    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"

 apply (rule iffI)

 apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])

 apply (drule distinct_remdups [THEN distinct_remdups

       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])

 apply simp

 done

 lemma multiset_of_compl_union [simp]:

   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"

   by (induct xs) (auto simp: add_ac)

 lemma count_multiset_of_length_filter:

   "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"

   by (induct xs) auto

 lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"

 apply (induct ls arbitrary: i)

  apply simp

 apply (case_tac i)

  apply auto

 done

 lemma multiset_of_remove1[simp]:

   "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"

 by (induct xs) (auto simp add: multiset_eq_iff)

 lemma multiset_of_eq_length:

   assumes "multiset_of xs = multiset_of ys"

   shows "length xs = length ys"

 using assms proof (induct xs arbitrary: ys)

   case Nil then show ?case by simp

 next

   case (Cons x xs)

   then have "x \<in># multiset_of ys" by (simp add: union_single_eq_member)

   then have "x \<in> set ys" by (simp add: in_multiset_in_set)

   from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)"

     by simp

   with Cons.hyps have "length xs = length (remove1 x ys)" .

   with `x \<in> set ys` show ?case

     by (auto simp add: length_remove1 dest: length_pos_if_in_set)

 qed

 lemma multiset_of_eq_length_filter:

   assumes "multiset_of xs = multiset_of ys"

   shows "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) ys)"

 proof (cases "z \<in># multiset_of xs")

   case False

   moreover have "\<not> z \<in># multiset_of ys" using assms False by simp

   ultimately show ?thesis by (simp add: count_multiset_of_length_filter)

 next

   case True

   moreover have "z \<in># multiset_of ys" using assms True by simp

   show ?thesis using assms proof (induct xs arbitrary: ys)

     case Nil then show ?case by simp

   next

     case (Cons x xs)

     from `multiset_of (x # xs) = multiset_of ys` [symmetric]

       have *: "multiset_of xs = multiset_of (remove1 x ys)"

       and "x \<in> set ys"

       by (auto simp add: mem_set_multiset_eq)

     from * have "length (filter (\<lambda>x. z = x) xs) = length (filter (\<lambda>y. z = y) (remove1 x ys))" by (rule Cons.hyps)

     moreover from `x \<in> set ys` have "length (filter (\<lambda>y. x = y) ys) > 0" by (simp add: filter_empty_conv)

     ultimately show ?case using `x \<in> set ys`

       by (simp add: filter_remove1) (auto simp add: length_remove1)

   qed

 qed

 context linorder

 begin

 lemma multiset_of_insort [simp]:

   "multiset_of (insort_key k x xs) = {#x#} + multiset_of xs"

   by (induct xs) (simp_all add: ac_simps)

 lemma multiset_of_sort [simp]:

   "multiset_of (sort_key k xs) = multiset_of xs"

   by (induct xs) (simp_all add: ac_simps)

 text {*

   This lemma shows which properties suffice to show that a function

   @{text "f"} with @{text "f xs = ys"} behaves like sort.

 *}

 lemma properties_for_sort_key:

   assumes "multiset_of ys = multiset_of 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> set ys"

     by (auto simp add: mem_set_multiset_eq intro!: ccontr)

   ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)

 qed

 lemma properties_for_sort:

   assumes multiset: "multiset_of ys = multiset_of xs"

   and "sorted ys"

   shows "sort xs = ys"

 proof (rule properties_for_sort_key)

   from multiset show "multiset_of ys = multiset_of xs" .

   from `sorted ys` show "sorted (map (\<lambda>x. x) ys)" by simp

   from multiset have "\<And>k. length (filter (\<lambda>y. k = y) ys) = length (filter (\<lambda>x. k = x) xs)"

     by (rule multiset_of_eq_length_filter)

   then have "\<And>k. replicate (length (filter (\<lambda>y. k = y) ys)) k = replicate (length (filter (\<lambda>x. k = x) xs)) k"

     by simp

   then show "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>y. k = y) ys = filter (\<lambda>x. k = x) xs"

     by (simp add: replicate_length_filter)

 qed

 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 "multiset_of ?rhs = multiset_of ?lhs"

     by (rule multiset_eqI) (auto simp add: multiset_of_filter)

 next

   show "sorted (map f ?rhs)"

     by (auto simp add: sorted_append intro: sorted_map_same)

 next

   fix l

   assume "l \<in> set ?rhs"

   let ?pivot = "f (xs ! (length xs div 2))"

   have *: "\<And>x. f l = f x \<longleftrightarrow> f x = f l" by auto

   have "[x \<leftarrow> sort_key f xs . f x = f l] = [x \<leftarrow> xs. f x = f l]"

     unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)

   with * have **: "[x \<leftarrow> sort_key f xs . f l = f x] = [x \<leftarrow> xs. f l = f x]" by simp

   have "\<And>x P. P (f x) ?pivot \<and> f l = f x \<longleftrightarrow> P (f l) ?pivot \<and> f l = f x" by auto

   then have "\<And>P. [x \<leftarrow> sort_key f xs . P (f x) ?pivot \<and> f l = f x] =

     [x \<leftarrow> sort_key f xs. P (f l) ?pivot \<and> f l = f x]" by simp

   note *** = this [of "op <"] this [of "op >"] this [of "op ="]

   show "[x \<leftarrow> ?rhs. f l = f x] = [x \<leftarrow> ?lhs. f l = f x]"

   proof (cases "f l" ?pivot rule: linorder_cases)

     case less then moreover have "f l \<noteq> ?pivot" and "\<not> f l > ?pivot" by auto

     ultimately show ?thesis

       by (simp add: filter_sort [symmetric] ** ***)

   next

     case equal then show ?thesis

       by (simp add: * less_le)

   next

     case greater then moreover have "f l \<noteq> ?pivot" and "\<not> f l < ?pivot" by auto

     ultimately 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 {* A stable parametrized quicksort *}

 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 multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"

   by (induct xs) (auto intro: order_trans)

 lemma multiset_of_update:

   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"

 proof (induct ls arbitrary: i)

   case Nil then show ?case by simp

 next

   case (Cons x xs)

   show ?case

   proof (cases i)

     case 0 then show ?thesis by simp

   next

     case (Suc i')

     with Cons show ?thesis

       apply simp

       apply (subst add_assoc)

       apply (subst add_commute [of "{#v#}" "{#x#}"])

       apply (subst add_assoc [symmetric])

       apply simp

       apply (rule mset_le_multiset_union_diff_commute)

       apply (simp add: mset_le_single nth_mem_multiset_of)

       done

   qed

 qed

 lemma multiset_of_swap:

   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>

     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"

   by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)

 subsubsection {* Association lists -- including rudimentary code generation *}

 definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where

   "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"

 lemma count_of_multiset:

   "count_of xs \<in> multiset"

 proof -

   let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"

   have "?A \<subseteq> dom (map_of xs)"

   proof

     fix x

     assume "x \<in> ?A"

     then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp

     then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto

     then show "x \<in> dom (map_of xs)" by auto

   qed

   with finite_dom_map_of [of xs] have "finite ?A"

     by (auto intro: finite_subset)

   then show ?thesis

     by (simp add: count_of_def fun_eq_iff multiset_def)

 qed

 lemma count_simps [simp]:

   "count_of [] = (\<lambda>_. 0)"

   "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"

   by (simp_all add: count_of_def fun_eq_iff)

 lemma count_of_empty:

   "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"

   by (induct xs) (simp_all add: count_of_def)

 lemma count_of_filter:

   "count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"

   by (induct xs) auto

 definition Bag :: "('a \<times> nat) list \<Rightarrow> 'a multiset" where

   "Bag xs = Abs_multiset (count_of xs)"

 code_datatype Bag

 lemma count_Bag [simp, code]:

   "count (Bag xs) = count_of xs"

   by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)

 lemma Mempty_Bag [code]:

   "{#} = Bag []"

   by (simp add: multiset_eq_iff)

 lemma single_Bag [code]:

   "{#x#} = Bag [(x, 1)]"

   by (simp add: multiset_eq_iff)

 lemma filter_Bag [code]:

   "Multiset.filter P (Bag xs) = Bag (filter (P \<circ> fst) xs)"

   by (rule multiset_eqI) (simp add: count_of_filter)

 lemma mset_less_eq_Bag [code]:

   "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A x)"

     (is "?lhs \<longleftrightarrow> ?rhs")

 proof

   assume ?lhs then show ?rhs

     by (auto simp add: mset_le_def count_Bag)

 next

   assume ?rhs

   show ?lhs

   proof (rule mset_less_eqI)

     fix x

     from `?rhs` have "count_of xs x \<le> count A x"

       by (cases "x \<in> fst ` set xs") (auto simp add: count_of_empty)

     then show "count (Bag xs) x \<le> count A x"

       by (simp add: mset_le_def count_Bag)

   qed

 qed

 instantiation multiset :: (equal) equal

 begin

 definition

   "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"

 instance proof

 qed (simp add: equal_multiset_def eq_iff)

 end

 lemma [code nbe]:

   "HOL.equal (A :: 'a::equal multiset) A \<longleftrightarrow> True"

   by (fact equal_refl)

 definition (in term_syntax)

   bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term)

     \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

   [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"

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

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

 instantiation multiset :: (random) random

 begin

 definition

   "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"

 instance ..

 end

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

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

 hide_const (open) bagify

 subsection {* The multiset order *}

 subsubsection {* Well-foundedness *}

 definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where

   "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>

       (\<forall>b. b :# K --> (b, a) \<in> r)}"

 definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where

   "mult r = (mult1 r)\<^sup>+"

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

 by (simp add: mult1_def)

 lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>

     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>

     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"

   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")

 proof (unfold mult1_def)

   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"

   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"

   let ?case1 = "?case1 {(N, M). ?R N M}"

   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"

   then have "\<exists>a' M0' K.

       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp

   then show "?case1 \<or> ?case2"

   proof (elim exE conjE)

     fix a' M0' K

     assume N: "N = M0' + K" and r: "?r K a'"

     assume "M0 + {#a#} = M0' + {#a'#}"

     then have "M0 = M0' \<and> a = a' \<or>

         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"

       by (simp only: add_eq_conv_ex)

     then show ?thesis

     proof (elim disjE conjE exE)

       assume "M0 = M0'" "a = a'"

       with N r have "?r K a \<and> N = M0 + K" by simp

       then have ?case2 .. then show ?thesis ..

     next

       fix K'

       assume "M0' = K' + {#a#}"

       with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)

       assume "M0 = K' + {#a'#}"

       with r have "?R (K' + K) M0" by blast

       with n have ?case1 by simp then show ?thesis ..

     qed

   qed

 qed

 lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"

 proof

   let ?R = "mult1 r"

   let ?W = "acc ?R"

   {

     fix M M0 a

     assume M0: "M0 \<in> ?W"

       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"

       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"

     have "M0 + {#a#} \<in> ?W"

     proof (rule accI [of "M0 + {#a#}"])

       fix N

       assume "(N, M0 + {#a#}) \<in> ?R"

       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>

           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"

         by (rule less_add)

       then show "N \<in> ?W"

       proof (elim exE disjE conjE)

         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"

         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..

         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..

         then show "N \<in> ?W" by (simp only: N)

       next

         fix K

         assume N: "N = M0 + K"

         assume "\<forall>b. b :# K --> (b, a) \<in> r"

         then have "M0 + K \<in> ?W"

         proof (induct K)

           case empty

           from M0 show "M0 + {#} \<in> ?W" by simp

         next

           case (add K x)

           from add.prems have "(x, a) \<in> r" by simp

           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast

           moreover from add have "M0 + K \<in> ?W" by simp

           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..

           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: add_assoc)

         qed

         then show "N \<in> ?W" by (simp only: N)

       qed

     qed

   } note tedious_reasoning = this

   assume wf: "wf r"

   fix M

   show "M \<in> ?W"

   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 wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"

     proof induct

       fix a

       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"

       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"

       proof

         fix M assume "M \<in> ?W"

         then show "M + {#a#} \<in> ?W"

           by (rule acc_induct) (rule tedious_reasoning [OF _ r])

       qed

     qed

     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..

   qed

 qed

 theorem wf_mult1: "wf r ==> wf (mult1 r)"

 by (rule acc_wfI) (rule all_accessible)

 theorem wf_mult: "wf r ==> wf (mult r)"

 unfolding mult_def by (rule wf_trancl) (rule wf_mult1)

 subsubsection {* Closure-free presentation *}

 text {* One direction. *}

 lemma mult_implies_one_step:

   "trans r ==> (M, N) \<in> mult r ==>

     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>

     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"

 apply (unfold mult_def mult1_def set_of_def)

 apply (erule converse_trancl_induct, clarify)

  apply (rule_tac x = M0 in exI, simp, clarify)

 apply (case_tac "a :# K")

  apply (rule_tac x = I in exI)

  apply (simp (no_asm))

  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)

  apply (simp (no_asm_simp) add: add_assoc [symmetric])

  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)

  apply (simp add: diff_union_single_conv)

  apply (simp (no_asm_use) add: trans_def)

  apply blast

 apply (subgoal_tac "a :# I")

  apply (rule_tac x = "I - {#a#}" in exI)

  apply (rule_tac x = "J + {#a#}" in exI)

  apply (rule_tac x = "K + Ka" in exI)

  apply (rule conjI)

   apply (simp add: multiset_eq_iff split: nat_diff_split)

  apply (rule conjI)

   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)

   apply (simp add: multiset_eq_iff split: nat_diff_split)

  apply (simp (no_asm_use) add: trans_def)

  apply blast

 apply (subgoal_tac "a :# (M0 + {#a#})")

  apply simp

 apply (simp (no_asm))

 done

 lemma one_step_implies_mult_aux:

   "trans r ==>

     \<forall>I J K. (size J = n \<and> J \<noteq> {#} \<and> (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r))

       --> (I + K, I + J) \<in> mult r"

 apply (induct_tac n, auto)

 apply (frule size_eq_Suc_imp_eq_union, clarify)

 apply (rename_tac "J'", simp)

 apply (erule notE, auto)

 apply (case_tac "J' = {#}")

  apply (simp add: mult_def)

  apply (rule r_into_trancl)

  apply (simp add: mult1_def set_of_def, blast)

 txt {* Now we know @{term "J' \<noteq> {#}"}. *}

 apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)

 apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)

 apply (erule ssubst)

 apply (simp add: Ball_def, auto)

 apply (subgoal_tac

   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},

     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")

  prefer 2

  apply force

 apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)

 apply (erule trancl_trans)

 apply (rule r_into_trancl)

 apply (simp add: mult1_def set_of_def)

 apply (rule_tac x = a in exI)

 apply (rule_tac x = "I + J'" in exI)

 apply (simp add: add_ac)

 done

 lemma one_step_implies_mult:

   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r

     ==> (I + K, I + J) \<in> mult r"

 using one_step_implies_mult_aux by blast

 subsubsection {* Partial-order properties *}

 definition less_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where

   "M' <# M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"

 definition le_multiset :: "'a\<Colon>order multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<=#" 50) where

   "M' <=# M \<longleftrightarrow> M' <# M \<or> M' = M"

 notation (xsymbols) less_multiset (infix "\<subset>#" 50)

 notation (xsymbols) le_multiset (infix "\<subseteq>#" 50)

 interpretation multiset_order: order le_multiset less_multiset

 proof -

   have irrefl: "\<And>M :: 'a multiset. \<not> M \<subset># M"

   proof

     fix M :: "'a multiset"

     assume "M \<subset># M"

     then have MM: "(M, M) \<in> mult {(x, y). x < y}" by (simp add: less_multiset_def)

     have "trans {(x'::'a, x). x' < x}"

       by (rule transI) simp

     moreover note MM

     ultimately have "\<exists>I J K. M = I + J \<and> M = I + K

       \<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})"

       by (rule mult_implies_one_step)

     then obtain I J K where "M = I + J" and "M = I + K"

       and "J \<noteq> {#}" and "(\<forall>k\<in>set_of K. \<exists>j\<in>set_of J. (k, j) \<in> {(x, y). x < y})" by blast

     then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_of K. \<exists>j\<in>set_of K. k < j" by auto

     have "finite (set_of K)" by simp

     moreover note aux2

     ultimately have "set_of K = {}"

       by (induct rule: finite_induct) (auto intro: order_less_trans)

     with aux1 show False by simp

   qed

   have trans: "\<And>K M N :: 'a multiset. K \<subset># M \<Longrightarrow> M \<subset># N \<Longrightarrow> K \<subset># N"

     unfolding less_multiset_def mult_def by (blast intro: trancl_trans)

   show "class.order (le_multiset :: 'a multiset \<Rightarrow> _) less_multiset" proof

   qed (auto simp add: le_multiset_def irrefl dest: trans)

 qed

 lemma mult_less_irrefl [elim!]:

   "M \<subset># (M::'a::order multiset) ==> R"

   by (simp add: multiset_order.less_irrefl)

 subsubsection {* Monotonicity of multiset union *}

 lemma mult1_union:

   "(B, D) \<in> mult1 r ==> (C + B, C + D) \<in> mult1 r"

 apply (unfold mult1_def)

 apply auto

 apply (rule_tac x = a in exI)

 apply (rule_tac x = "C + M0" in exI)

 apply (simp add: add_assoc)

 done

 lemma union_less_mono2: "B \<subset># D ==> C + B \<subset># C + (D::'a::order multiset)"

 apply (unfold less_multiset_def mult_def)

 apply (erule trancl_induct)

  apply (blast intro: mult1_union)

 apply (blast intro: mult1_union trancl_trans)

 done

 lemma union_less_mono1: "B \<subset># D ==> B + C \<subset># D + (C::'a::order multiset)"

 apply (subst add_commute [of B C])

 apply (subst add_commute [of D C])

 apply (erule union_less_mono2)

 done

 lemma union_less_mono:

   "A \<subset># C ==> B \<subset># D ==> A + B \<subset># C + (D::'a::order multiset)"

   by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)

 interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset

 proof

 qed (auto simp add: le_multiset_def intro: union_less_mono2)

 subsection {* The fold combinator *}

 text {*

   The intended behaviour is

   @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}

   if @{text f} is associative-commutative. 

 *}

 text {*

   The graph of @{text "fold_mset"}, @{text "z"}: the start element,

   @{text "f"}: folding function, @{text "A"}: the multiset, @{text

   "y"}: the result.

 *}

 inductive 

   fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 

   for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 

   and z :: 'b

 where

   emptyI [intro]:  "fold_msetG f z {#} z"

 | insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"

 inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"

 inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 

 definition

   fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where

   "fold_mset f z A = (THE x. fold_msetG f z A x)"

 lemma Diff1_fold_msetG:

   "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"

 apply (frule_tac x = x in fold_msetG.insertI)

 apply auto

 done

 lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"

 apply (induct A)

  apply blast

 apply clarsimp

 apply (drule_tac x = x in fold_msetG.insertI)

 apply auto

 done

 lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"

 unfolding fold_mset_def by blast

 context fun_left_comm

 begin

 lemma fold_msetG_determ:

   "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"

 proof (induct arbitrary: x y z rule: full_multiset_induct)

   case (less M x\<^isub>1 x\<^isub>2 Z)

   have IH: "\<forall>A. A < M \<longrightarrow> 

     (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'

                \<longrightarrow> x' = x)" by fact

   have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+

   show ?case

   proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])

     assume "M = {#}" and "x\<^isub>1 = Z"

     then show ?case using Mfoldx\<^isub>2 by auto 

   next

     fix B b u

     assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"

     then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto

     show ?case

     proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])

       assume "M = {#}" "x\<^isub>2 = Z"

       then show ?case using Mfoldx\<^isub>1 by auto

     next

       fix C c v

       assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"

       then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto

       then have CsubM: "C < M" by simp

       from MBb have BsubM: "B < M" by simp

       show ?case

       proof cases

         assume "b=c"

         then moreover have "B = C" using MBb MCc by auto

         ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto

       next

         assume diff: "b \<noteq> c"

         let ?D = "B - {#c#}"

         have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff

           by (auto intro: insert_noteq_member dest: sym)

         have "B - {#c#} < B" using cinB by (rule mset_less_diff_self)

         then have DsubM: "?D < M" using BsubM by (blast intro: order_less_trans)

         from MBb MCc have "B + {#b#} = C + {#c#}" by blast

         then have [simp]: "B + {#b#} - {#c#} = C"

           using MBb MCc binC cinB by auto

         have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"

           using MBb MCc diff binC cinB

           by (auto simp: multiset_add_sub_el_shuffle)

         then obtain d where Dfoldd: "fold_msetG f Z ?D d"

           using fold_msetG_nonempty by iprover

         then have "fold_msetG f Z B (f c d)" using cinB

           by (rule Diff1_fold_msetG)

         then have "f c d = u" using IH BsubM Bu by blast

         moreover 

         have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd

           by (auto simp: multiset_add_sub_el_shuffle 

             dest: fold_msetG.insertI [where x=b])

         then have "f b d = v" using IH CsubM Cv by blast

         ultimately show ?thesis using x\<^isub>1 x\<^isub>2

           by (auto simp: fun_left_comm)

       qed

     qed

   qed

 qed

 lemma fold_mset_insert_aux:

   "(fold_msetG f z (A + {#x#}) v) =

     (\<exists>y. fold_msetG f z A y \<and> v = f x y)"

 apply (rule iffI)

  prefer 2

  apply blast

 apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])

 apply (blast intro: fold_msetG_determ)

 done

 lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"

 unfolding fold_mset_def by (blast intro: fold_msetG_determ)

 lemma fold_mset_insert:

   "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"

 apply (simp add: fold_mset_def fold_mset_insert_aux)

 apply (rule the_equality)

  apply (auto cong add: conj_cong 

      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)

 done

 lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"

 by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x])

 lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"

 using fold_mset_insert [of z "{#}"] by simp

 lemma fold_mset_union [simp]:

   "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"

 proof (induct A)

   case empty then show ?case by simp

 next

   case (add A x)

   have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac)

   then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" 

     by (simp add: fold_mset_insert)

   also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"

     by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)

   finally show ?case .

 qed

 lemma fold_mset_fusion:

   assumes "fun_left_comm g"

   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")

 proof -

   interpret fun_left_comm g by (fact assms)

   show "PROP ?P" by (induct A) auto

 qed

 lemma fold_mset_rec:

   assumes "a \<in># A" 

   shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"

 proof -

   from assms obtain A' where "A = A' + {#a#}"

     by (blast dest: multi_member_split)

   then show ?thesis by simp

 qed

 end

 text {*

   A note on code generation: When defining some function containing a

   subterm @{term"fold_mset F"}, code generation is not automatic. When

   interpreting locale @{text left_commutative} with @{text F}, the

   would be code thms for @{const fold_mset} become thms like

   @{term"fold_mset F z {#} = z"} where @{text F} 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 @{text

   F}. See the image operator below.

 *}

 subsection {* Image *}

 definition image_mset :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a multiset \<Rightarrow> 'b multiset" where

   "image_mset f = fold_mset (op + o single o f) {#}"

 interpretation image_left_comm: fun_left_comm "op + o single o f"

 proof qed (simp add: add_ac fun_eq_iff)

 lemma image_mset_empty [simp]: "image_mset f {#} = {#}"

 by (simp add: image_mset_def)

 lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"

 by (simp add: image_mset_def)

 lemma image_mset_insert:

   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"

 by (simp add: image_mset_def add_ac)

 lemma image_mset_union [simp]:

   "image_mset f (M+N) = image_mset f M + image_mset f N"

 apply (induct N)

  apply simp

 apply (simp add: add_assoc [symmetric] image_mset_insert)

 done

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

 by (induct M) simp_all

 lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"

 by (cases M) auto

 syntax

   "_comprehension1_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"

       ("({#_/. _ :# _#})")

 translations

   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"

 syntax

   "_comprehension2_mset" :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"

       ("({#_/ | _ :# _./ _#})")

 translations

   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"

 text {*

   This allows to write not just filters like @{term "{#x:#M. x<c#}"}

   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]

   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as

   @{term "{#x+x|x:#M. x<c#}"}.

 *}

 enriched_type 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

 next

   show "image_mset id = id"

   proof

     fix A

     show "image_mset id A = id A"

       by (induct A) simp_all

   qed

 qed

 subsection {* Termination proofs with multiset orders *}

 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_of A, set_of 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_of A, set_of 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_of A, set_of 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_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" 

     by auto

   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less" 

     unfolding pair_leq_def by auto

   thus ?case

   proof

     assume [simp]: "x = y"

     have

       "{#x#} + X = A + ({#y#}+Z) 

       \<and> {#y#} + Y = B + ({#y#}+Z)

       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"

       by (auto simp: add_ac)

     thus ?case by (intro exI)

   next

     assume A: "(x, y) \<in> pair_less"

     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"

     have "{#x#} + X = ?A' + Z"

       "{#y#} + Y = ?B' + Z"

       by (auto simp add: add_ac)

     moreover have 

       "(set_of ?A', set_of ?B') \<in> max_strict"

       using 1 A 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_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"

   and   ms_weakI1:  "(set_of A, set_of 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_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"

     by blast

   {

     assume max: "(set_of A, set_of B) \<in> max_strict"

     from mx_or_empty

     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"

     proof

       assume max': "(set_of A', set_of B') \<in> max_strict"

       with max have "(set_of (A + A'), set_of (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:add_ac)

     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:add_ac)

 qed

 lemma empty_neutral: "{#} + x = x" "x + {#} = x"

 and nonempty_plus: "{# x #} + rs \<noteq> {#}"

 and nonempty_single: "{# x #} \<noteq> {#}"

 by auto

 setup {*

 let

   fun msetT T = Type (@{type_name multiset}, [T]);

   fun mk_mset T [] = Const (@{const_abbrev Mempty}, msetT T)

     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x

     | mk_mset T (x :: xs) =

           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $

                 mk_mset T [x] $ mk_mset T xs

   fun mset_member_tac m i =

       (if m <= 0 then

            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i

        else

            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)

   val mset_nonempty_tac =

       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}

   val regroup_munion_conv =

       Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}

         (map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_neutral}))

   fun unfold_pwleq_tac i =

     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))

       ORELSE (rtac @{thm pw_leq_lstep} i)

       ORELSE (rtac @{thm pw_leq_empty} i)

   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_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_of_simps,

     smsI'= @{thm ms_strictI}, wmsI2''= @{thm ms_weakI2}, wmsI1= @{thm ms_weakI1},

     reduction_pair= @{thm ms_reduction_pair}

   })

 end

 *}

 subsection {* Legacy theorem bindings *}

 lemmas multi_count_eq = multiset_eq_iff [symmetric]

 lemma union_commute: "M + N = N + (M::'a multiset)"

   by (fact add_commute)

 lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"

   by (fact add_assoc)

 lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"

   by (fact add_left_commute)

 lemmas union_ac = union_assoc union_commute union_lcomm

 lemma union_right_cancel: "M + K = N + K \<longleftrightarrow> M = (N::'a multiset)"

   by (fact add_right_cancel)

 lemma union_left_cancel: "K + M = K + N \<longleftrightarrow> M = (N::'a multiset)"

   by (fact add_left_cancel)

 lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"

   by (fact add_imp_eq)

 lemma mset_less_trans: "(M::'a multiset) < K \<Longrightarrow> K < N \<Longrightarrow> M < N"

   by (fact order_less_trans)

 lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"

   by (fact inf.commute)

 lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"

   by (fact inf.assoc [symmetric])

 lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"

   by (fact inf.left_commute)

 lemmas multiset_inter_ac =

   multiset_inter_commute

   multiset_inter_assoc

   multiset_inter_left_commute

 lemma mult_less_not_refl:

   "\<not> M \<subset># (M::'a::order multiset)"

   by (fact multiset_order.less_irrefl)

 lemma mult_less_trans:

   "K \<subset># M ==> M \<subset># N ==> K \<subset># (N::'a::order multiset)"

   by (fact multiset_order.less_trans)

 lemma mult_less_not_sym:

   "M \<subset># N ==> \<not> N \<subset># (M::'a::order multiset)"

   by (fact multiset_order.less_not_sym)

 lemma mult_less_asym:

   "M \<subset># N ==> (\<not> P ==> N \<subset># (M::'a::order multiset)) ==> P"

   by (fact multiset_order.less_asym)

 ML {*

 fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))

                       (Const _ $ t') =

     let

       val (maybe_opt, ps) =

         Nitpick_Model.dest_plain_fun t' ||> op ~~

         ||> map (apsnd (snd o HOLogic.dest_number))

       fun elems_for t =

         case AList.lookup (op =) ps t of

           SOME n => replicate n t

         | NONE => [Const (maybe_name, elem_T --> elem_T) $ t]

     in

       case maps elems_for (all_values elem_T) @

            (if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]

             else []) of

         [] => Const (@{const_name zero_class.zero}, T)

       | ts => foldl1 (fn (t1, t2) =>

                          Const (@{const_name plus_class.plus}, T --> T --> T)

                          $ t1 $ t2)

                      (map (curry (op $) (Const (@{const_name single},

                                                 elem_T --> T))) ts)

     end

   | multiset_postproc _ _ _ _ t = t

 *}

 declaration {*

 Nitpick_Model.register_term_postprocessor @{typ "'a multiset"}

     multiset_postproc

 *}

 end