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products/Sources/formale Sprachen/Isabelle/HOL/Library/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 188 kB

Quelle Multiset.thy Sprache: Isabelle

(*  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 = \{f :: 'a \ nat. finite {x. f x > 0}}\
  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 \ (\a. count M a = count N a)"
  by (simp only: count_inject [symmetric] fun_eq_iff)

lemma multiset_eqI: "(\x. count A x = count B x) \ 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 \ bool" where
  [code_abbrev]: "is_empty A \ 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 \ 'a multiset \ 'a multiset" is
  "\a M b. if b = a then Suc (M b) else M b"
by (rule add_mset_in_multiset)

syntax
  "_multiset" :: "args \ 'a multiset" (\(\indent=2 notation=\mixfix multiset enumeration\\{#_#})\)
syntax_consts
  "_multiset" \<rightleftharpoons> add_mset
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]: "{#} \ 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 \ 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>(\<open>notation=\<open>infix \<in>#\<close>\<close>_/ \<in># _)\<close> [50, 51] 50)

notation  (ASCII)
  member_mset  (\<open>'(:#')\<close>) and
  member_mset  (\<open>(\<open>notation=\<open>infix :#\<close>\<close>_/ :# _)\<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>(\<open>notation=\<open>infix \<notin>#\<close>\<close>_/ \<notin># _)\<close> [50, 51] 50)

notation  (ASCII)
  not_member_mset  (\<open>'(~:#')\<close>) and
  not_member_mset  (\<open>(\<open>notation=\<open>infix ~:#\<close>\<close>_/ ~:# _)\<close> [50, 51] 50)

context
begin

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

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

end

syntax
  "_MBall"       :: "pttrn \ 'a set \ bool \ bool"
    (\<open>(\<open>indent=3 notation=\<open>binder \<forall>\<close>\<close>\<forall>_\<in>#_./ _)\<close> [0, 0, 10] 10)
  "_MBex"        :: "pttrn \ 'a set \ bool \ bool"
    (\<open>(\<open>indent=3 notation=\<open>binder \<exists>\<close>\<close>\<exists>_\<in>#_./ _)\<close> [0, 0, 10] 10)
syntax  (ASCII)
  "_MBall"       :: "pttrn \ 'a set \ bool \ bool"
    (\<open>(\<open>indent=3 notation=\<open>binder \<forall>\<close>\<close>\<forall>_:#_./ _)\<close> [0, 0, 10] 10)
  "_MBex"        :: "pttrn \ 'a set \ bool \ bool"
    (\<open>(\<open>indent=3 notation=\<open>binder \<exists>\<close>\<close>\<exists>_:#_./ _)\<close> [0, 0, 10] 10)
syntax_consts
  "_MBall" \<rightleftharpoons> Multiset.Ball and
  "_MBex" \<rightleftharpoons> Multiset.Bex
translations
  "\x\#A. P" \ "CONST Multiset.Ball A (\x. P)"
  "\x\#A. P" \ "CONST Multiset.Bex A (\x. P)"

typed_print_translation \<open>
[(\<^const_syntax>\<open>Multiset.Ball\<close>, Syntax_Trans.preserve_binder_abs2_tr' \<^syntax_const>\<open>_MBall\<close>),
  (\<^const_syntax>\<open>Multiset.Bex\<close>, Syntax_Trans.preserve_binder_abs2_tr' \<^syntax_const>\<open>_MBex\<close>)]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>

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

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

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

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

lemma in_countE:
  assumes "x \# 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 \ Suc 0 \ x \# M"
  by (simp add: Suc_le_eq)

lemma count_greater_eq_one_iff [simp]:
  "count M x \ 1 \ x \# 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 = {} \ 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 \ {#}"
  obtains x where "x \# A"
proof -
  have "\x. x \# A" by (rule ccontr) (insert assms, auto)
  with that show ?thesis by blast
qed

lemma count_gt_imp_in_mset: "count M x > n \ x \# M"
  using count_greater_zero_iff by fastforce

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 \ 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)

(* TODO: reverse arguments to prevent unfolding loop *)
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 \# M - N \ count N a < count M a"
  by (simp add: set_mset_def)

lemma count_in_diffI:
  assumes "\n. count N x = n + count M x \ False"
  shows "x \# M - N"
proof (rule ccontr)
  assume "x \# 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 \# 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 \# M - N"
  shows "a \# M"
proof -
  have "0 \ 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 \ {#} - 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 \# M \ add_mset x (M - {#x#}) = M"
  by (clarsimp simp: multiset_eq_iff)

lemma insert_DiffM2: "x \# M \ (M - {#x#}) + {#x#} = M"
  by simp

lemma diff_union_swap: "a \ b \ add_mset b (M - {#a#}) = add_mset b M - {#a#}"
  by (auto simp add: multiset_eq_iff)

lemma diff_add_mset_swap [simp]: "b \# A \ 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 \# M \ 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 \# J \ I + J - {#a#} = I + (J - {#a#})"
  by (simp add: multiset_eq_iff Suc_le_eq)

lemma mset_add [elim?]:
  assumes "a \# 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 \# A + B \ a \# A \ a \# B"
  by auto

lemma count_minus_inter_lt_count_minus_inter_iff:
  "count (M2 - M1) y < count (M1 - M2) y \ y \# M1 - M2"
  by (meson count_greater_zero_iff gr_implies_not_zero in_diff_count leI order.strict_trans2
      order_less_asym)

lemma minus_inter_eq_minus_inter_iff:
  "(M1 - M2) = (M2 - M1) \ set_mset (M1 - M2) = set_mset (M2 - M1)"
  by (metis add.commute count_diff count_eq_zero_iff diff_add_zero in_diff_countE multiset_eq_iff)

subsubsection \<open>Min and Max\<close>

abbreviation Min_mset :: "'a::linorder multiset \ 'a" where
"Min_mset m \ Min (set_mset m)"

abbreviation Max_mset :: "'a::linorder multiset \ 'a" where
"Max_mset m \ Max (set_mset m)"

lemma
  Min_in_mset: "M \ {#} \ Min_mset M \# M" and
  Max_in_mset: "M \ {#} \ Max_mset M \# M"
  by simp+

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

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

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

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

lemma multi_self_add_other_not_self [simp]: "M = add_mset x M \ 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: "\ x \# M \ M - {#x#} = M"
  by (auto simp add: multiset_eq_iff not_in_iff)

lemma diff_single_eq_union: "x \# M \ M - {#x#} = N \ M = add_mset x N"
  by auto

lemma union_single_eq_diff: "add_mset x M = N \ 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 \ x \# N"
  by auto

lemma add_mset_remove_trivial_If:
  "add_mset a (N - {#a#}) = (if a \# 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#} \ M = {#a#} \ N = {#} \ M = {#} \ 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 \ {#a#} = M \ N = {#} \ M = {#} \ {#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 \ M = N \ a = b \ M = add_mset b (N - {#a#}) \ N = add_mset a (M - {#b#})"
  (is "?lhs \ ?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 \# 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#} \ b = a \ M = {#}"
  by (auto simp: add_eq_conv_diff)

lemma single_eq_add_mset [iff]: "{#a#} = add_mset b M \ b = a \ 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 \ c"
  shows "c \# B"
proof -
  have "c \# add_mset c C" by simp
  have nc: "\ c \# {#b#}" using bnotc by simp
  then have "c \# add_mset b B" using BC by simp
  then show "c \# 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 \# M \ \A. M = add_mset x A"
  by (rule exI [where x = "M - {#x#}"]) simp

lemma multiset_add_sub_el_shuffle:
  assumes "c \# B"
    and "b \ 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#} \ M = {#} \ x = y"
  by auto

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

definition subseteq_mset :: "'a multiset \ 'a multiset \ bool" (infix \\#\ 50)
  where "A \# B \ (\a. count A a \ count B a)"

definition subset_mset :: "'a multiset \ 'a multiset \ bool" (infix \\#\ 50)
  where "A \# B \ A \# B \ A \ B"

abbreviation (input) supseteq_mset :: "'a multiset \ 'a multiset \ bool" (infix \\#\ 50)
  where "supseteq_mset A B \ B \# A"

abbreviation (input) supset_mset :: "'a multiset \ 'a multiset \ bool" (infix \\#\ 50)
  where "supset_mset A B \ B \# A"

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

notation (ASCII)
  subseteq_mset  (infix \<open><=#\<close> 50) and
  subset_mset  (infix \<open><#\<close> 50) and
  supseteq_mset  (infix \<open>>=#\<close> 50) and
  supset_mset  (infix \<open>>#\<close> 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 "(-)" "(\#)" "(\#)"
  by standard
    \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>

lemma mset_subset_eqI:
  "(\a. count A a \ count B a) \ A \# B"
  by (simp add: subseteq_mset_def)

lemma mset_subset_eq_count:
  "A \# B \ count A a \ count B a"
  by (simp add: subseteq_mset_def)

lemma mset_subset_eq_exists_conv: "(A::'a multiset) \# B \ (\C. B = A + C)"
  unfolding subseteq_mset_def
  by (metis add_diff_cancel_left' count_diff count_union le_Suc_ex le_add_same_cancel1 multiset_eq_iff zero_le)

interpretation subset_mset: ordered_cancel_comm_monoid_diff "(+)" 0 "(\#)" "(\#)" "(-)"
  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 \# B + C \ A \# B"
   by (fact subset_mset.add_le_cancel_right)

lemma mset_subset_eq_mono_add_left_cancel: "C + (A::'a multiset) \# C + B \ A \# B"
   by (fact subset_mset.add_le_cancel_left)

lemma mset_subset_eq_mono_add: "(A::'a multiset) \# B \ C \# D \ A + C \# B + D"
   by (fact subset_mset.add_mono)

lemma mset_subset_eq_add_left: "(A::'a multiset) \# A + B"
   by simp

lemma mset_subset_eq_add_right: "B \# (A::'a multiset) + B"
   by simp

lemma single_subset_iff [simp]:
  "{#a#} \# M \ a \# M"
  by (auto simp add: subseteq_mset_def Suc_le_eq)

lemma mset_subset_eq_single: "a \# B \ {#a#} \# 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 \# B \ 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 \# A \ A - B + C = A + C - B"
  by (fact subset_mset.add_diff_assoc2)

lemma diff_subset_eq_self[simp]:
  "(M::'a multiset) - N \# M"
  by (simp add: subseteq_mset_def)

lemma mset_subset_eqD:
  assumes "A \# B" and "x \# A"
  shows "x \# 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 \# B \ x \# A \ x \# B"
  by (auto intro: mset_subset_eqD [of A])

lemma set_mset_mono:
  "A \# B \ set_mset A \ set_mset B"
  by (metis mset_subset_eqD subsetI)

lemma mset_subset_eq_insertD:
  assumes "add_mset x A \# B"
  shows "x \# B \ A \# B"
proof
  show "x \# B"
    using assms by (simp add: mset_subset_eqD)
  have "A \# add_mset x A"
    by (metis (no_types) add_mset_add_single mset_subset_eq_add_left)
  then have "A \# add_mset x A"
    by (meson multi_self_add_other_not_self subset_mset.le_imp_less_or_eq)
  then show "A \# B"
    using assms subset_mset.strict_trans2 by blast
qed

lemma mset_subset_insertD:
  "add_mset x A \# B \ x \# B \ A \# B"
  by (rule mset_subset_eq_insertD) simp

lemma mset_subset_of_empty[simp]: "A \# {#} \ False"
  by (simp only: subset_mset.not_less_zero)

lemma empty_subset_add_mset[simp]: "{#} \# add_mset x M"
  by (auto intro: subset_mset.gr_zeroI)

lemma empty_le: "{#} \# A"
  by (fact subset_mset.zero_le)

lemma insert_subset_eq_iff:
  "add_mset a A \# B \ a \# B \ A \# B - {#a#}"
  using mset_subset_eq_insertD subset_mset.le_diff_conv2 by fastforce

lemma insert_union_subset_iff:
  "add_mset a A \# B \ a \# B \ A \# B - {#a#}"
  by (auto simp add: insert_subset_eq_iff subset_mset_def)

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

lemma multi_psub_of_add_self [simp]: "A \# add_mset x A"
  by (auto simp: subset_mset_def subseteq_mset_def)

lemma multi_psub_self: "A \# A = False"
  by simp

lemma mset_subset_add_mset [simp]: "add_mset x N \# add_mset x M \ N \# 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 \# B \ B - {#c#} \# B"
  by (auto simp: subset_mset_def elim: mset_add)

lemma Diff_eq_empty_iff_mset: "A - B = {#} \ A \# B"
  by (auto simp: multiset_eq_iff subseteq_mset_def)

lemma add_mset_subseteq_single_iff[iff]: "add_mset a M \# {#b#} \ M = {#} \ a = b"
proof
  assume A: "add_mset a M \# {#b#}"
  then have \<open>a = b\<close>
    by (auto dest: mset_subset_eq_insertD)
  then show "M={#} \ a=b"
    using A by (simp add: mset_subset_eq_add_mset_cancel)
qed simp

lemma nonempty_subseteq_mset_eq_single: "M \ {#} \ M \# {#x#} \ M = {#x#}"
  by (cases M) (metis single_is_union subset_mset.less_eqE)

lemma nonempty_subseteq_mset_iff_single: "(M \ {#} \ M \# {#x#} \ P) \ M = {#x#} \ P"
  by (cases M) (metis empty_not_add_mset nonempty_subseteq_mset_eq_single subset_mset.order_refl)

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>
proof
  show "\a b. (b \# a) = (a = a \# b)"
    by (simp add: Diff_eq_empty_iff_mset union_mset_def)
  show "\a b. (b \# a) = (a = a \# b \ a \ b)"
    by (metis Diff_eq_empty_iff_mset add_cancel_left_right subset_mset_def union_mset_def)
qed (auto simp: multiset_eqI union_mset_def)

interpretation subset_mset: semilattice_sup \<open>(\<union>#)\<close> \<open>(\<subseteq>#)\<close> \<open>(\<subset>#)\<close>
proof -
  have [simp]: "m \ n \ q \ n \ m + (q - m) \ n" for m n q :: nat
    by arith
  show "class.semilattice_sup (\#) (\#) (\#)"
    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 "(\#)" "(\#)" "(\#)"
  "(\#)" "{#}"
  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 \# B) = set_mset A \ set_mset B"
  by (simp only: set_mset_def) auto

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

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

lemma multiset_inter_single[simp]: "a \ b \ {#a#} \# {#b#} = {#}"
  by (rule multiset_eqI) auto

lemma multiset_union_diff_commute:
  assumes "B \# 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 \ 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 \# B = {#} \ (\a. a \# A \ a \# B)"
  by (metis disjoint_iff set_mset_eq_empty_iff set_mset_inter)

lemma inter_mset_empty_distrib_right: "A \# (B + C) = {#} \ A \# B = {#} \ A \# C = {#}"
  by (meson disjunct_not_in union_iff)

lemma inter_mset_empty_distrib_left: "(A + B) \# C = {#} \ A \# C = {#} \ B \# C = {#}"
  by (meson disjunct_not_in union_iff)

lemma add_mset_inter_add_mset [simp]:
  "add_mset a A \# add_mset a B = add_mset a (A \# B)"
  by (rule multiset_eqI) simp

lemma add_mset_disjoint [simp]:
  "add_mset a A \# B = {#} \ a \# B \ A \# B = {#}"
  "{#} = add_mset a A \# B \ a \# B \ {#} = A \# B"
  by (auto simp: disjunct_not_in)

lemma disjoint_add_mset [simp]:
  "B \# add_mset a A = {#} \ a \# B \ B \# A = {#}"
  "{#} = A \# add_mset b B \ b \# A \ {#} = A \# B"
  by (auto simp: disjunct_not_in)

lemma inter_add_left1: "\ x \# N \ (add_mset x M) \# N = M \# N"
  by (simp add: multiset_eq_iff not_in_iff)

lemma inter_add_left2: "x \# N \ (add_mset x M) \# N = add_mset x (M \# (N - {#x#}))"
  by (auto simp add: multiset_eq_iff elim: mset_add)

lemma inter_add_right1: "\ x \# N \ N \# (add_mset x M) = N \# M"
  by (simp add: multiset_eq_iff not_in_iff)

lemma inter_add_right2: "x \# N \ N \# (add_mset x M) = add_mset x ((N - {#x#}) \# M)"
  by (auto simp add: multiset_eq_iff elim: mset_add)

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

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

lemma inter_iff:
  "a \# A \# B \ a \# A \ a \# B"
  by simp

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

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

lemma inter_subset_eq_union:
  "A \# B \# 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]: "\ x \# N \ (add_mset x M) \# N = add_mset x (M \# N)"
  by (simp add: multiset_eq_iff not_in_iff)

lemma sup_union_left2: "x \# N \ (add_mset x M) \# N = add_mset x (M \# (N - {#x#}))"
  by (simp add: multiset_eq_iff)

lemma sup_union_right1 [simp]: "\ x \# N \ N \# (add_mset x M) = add_mset x (N \# M)"
  by (simp add: multiset_eq_iff not_in_iff)

lemma sup_union_right2: "x \# N \ N \# (add_mset x M) = add_mset x ((N - {#x#}) \# M)"
  by (simp add: multiset_eq_iff)

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

lemma union_sup_distrib_right:
  "C + A \# B = (C + A) \# (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 \# B = A \# B"
  by (auto simp add: multiset_eq_iff)

lemma union_diff_sup_eq_inter:
  "A + B - A \# B = A \# 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 \ 'a \ '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': \repeat_mset (i - j) u = repeat_mset i u - repeat_mset j u\
  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

lemma set_mset_sum: "finite A \ set_mset (\x\A. f x) = (\x\A. set_mset (f x))"
  by (induction A rule: finite_induct) auto

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 \ (i::nat) \ (repeat_mset i u + m \# repeat_mset j u + n) = (repeat_mset (i-j) u + m \# n)"
  by (auto simp add: subseteq_mset_def nat_le_add_iff1)

lemma mset_subseteq_add_iff2:
  "i \ (j::nat) \ (repeat_mset i u + m \# repeat_mset j u + n) = (m \# repeat_mset (j-i) u + n)"
  by (auto simp add: subseteq_mset_def nat_le_add_iff2)

lemma mset_subset_add_iff1:
  "j \ (i::nat) \ (repeat_mset i u + m \# repeat_mset j u + n) = (repeat_mset (i-j) u + m \# 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 \ (j::nat) \ (repeat_mset i u + m \# repeat_mset j u + n) = (m \# 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>K Cancel_Simprocs.eq_cancel\<close>

simproc_setup msetsubset_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>K Multiset_Simprocs.subset_cancel_msets\<close>

simproc_setup msetsubset_eq_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>K 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>K Cancel_Simprocs.diff_cancel\<close>

subsubsection \<open>Conditionally complete lattice\<close>

instantiation multiset :: (type) Inf
begin

lift_definition Inf_multiset :: "'a multiset set \ 'a multiset" is
  "\A i. if A = {} then 0 else Inf ((\f. f i) ` A)"
proof -
  fix A :: "('a \ nat) set"
  assume *: "\f. f \ A \ 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 \ A" by blast
    hence "{i. Inf ((\f. f i) ` A) > 0} \ {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 ((\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 \ {} \ count (Inf A) x = Inf ((\X. count X x) ` A)"
  by transfer simp_all

instantiation multiset :: (type) Sup
begin

definition Sup_multiset :: "'a multiset set \ 'a multiset" where
  "Sup_multiset A = (if A \ {} \ 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: "\ subset_mset.bdd_above A \ 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 ((\X. count X x) ` A)"
proof -
  from assms obtain Y where Y: "\X\A. X \# Y"
    by (meson subset_mset.bdd_above.E)
  hence "count X x \ count Y x" if "X \ 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 \ {}" "subset_mset.bdd_above (A :: 'a multiset set)"
  shows   "finite (\X\A. {x. count X x > 0})"
proof -
  from assms obtain Y where Y: "\X\A. X \# Y"
    by (meson subset_mset.bdd_above.E)
  hence "count X x \ count Y x" if "X \ A" for X x
    using that by (auto intro: mset_subset_eq_count)
  hence "(\X\A. {x. count X x > 0}) \ {x. count Y x > 0}"
    by safe (erule less_le_trans)
  moreover have "finite \" 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 ((\X. count X i) ` A) > 0} \ (\X\A. {i. 0 < count X i})"
  proof safe
    fix i assume pos: "(SUP X\A. count X i) > 0"
    show "i \ (\X\A. {i. 0 < count X i})"
    proof (rule ccontr)
      assume "i \ (\X\A. {i. 0 < count X i})"
      hence "\X\A. count X i \ 0" by (auto simp: count_eq_zero_iff)
      with that have "(SUP X\A. count X i) \ 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 \"
    by (rule bdd_above_multiset_imp_finite_support)
  ultimately show "finite {i. Sup ((\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 "(\#)" "(\#)" "(\#)" "(\#)"
proof
  fix X :: "'a multiset" and A
  assume "X \ A"
  show "Inf A \# X"
    by (metis \<open>X \<in> A\<close> count_Inf_multiset_nonempty empty_iff image_eqI mset_subset_eqI wellorder_Inf_le1)
next
  fix X :: "'a multiset" and A
  assume nonempty: "A \ {}" and le: "\Y. Y \ A \ X \# Y"
  show "X \# Inf A"
  proof (rule mset_subset_eqI)
    fix x
    from nonempty have "count X x \ (INF X\A. count X x)"
      by (intro cInf_greatest) (auto intro: mset_subset_eq_count le)
    also from nonempty have "\ = count (Inf A) x" by (simp add: count_Inf_multiset_nonempty)
    finally show "count X x \ count (Inf A) x" .
  qed
next
  fix X :: "'a multiset" and A
  assume X: "X \ A" and bdd: "subset_mset.bdd_above A"
  show "X \# Sup A"
  proof (rule mset_subset_eqI)
    fix x
    from X have "A \ {}" by auto
    have "count X x \ (SUP X\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\A. count X x) = count (Sup A) x" by simp
    finally show "count X x \ count (Sup A) x" .
  qed
next
  fix X :: "'a multiset" and A
  assume nonempty: "A \ {}" and ge: "\Y. Y \ A \ Y \# X"
  from ge have bdd: "subset_mset.bdd_above A"
    by blast
  show "Sup A \# 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\A. count X x)" .
    also from nonempty have "\ \ count X x"
      by (intro cSup_least) (auto intro: mset_subset_eq_count ge)
    finally show "count (Sup A) x \ count X x" .
  qed
qed \<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>

lemma set_mset_Inf:
  assumes "A \ {}"
  shows   "set_mset (Inf A) = (\X\A. set_mset X)"
proof safe
  fix x X assume "x \# Inf A" "X \ A"
  hence nonempty: "A \ {}" 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 \# X" by simp
next
  fix x assume x: "x \ (\X\A. set_mset X)"
  hence "{#x#} \# X" if "X \ A" for X using that by auto
  from assms and this have "{#x#} \# Inf A" by (rule subset_mset.cInf_greatest)
  thus "x \# Inf A" by simp
qed

lemma in_Inf_multiset_iff:
  assumes "A \ {}"
  shows   "x \# Inf A \ (\X\A. x \# X)"
proof -
  from assms have "set_mset (Inf A) = (\X\A. set_mset X)" by (rule set_mset_Inf)
  also have "x \ \ \ (\X\A. x \# X)" by simp
  finally show ?thesis .
qed

lemma in_Inf_multisetD: "x \# Inf A \ X \ A \ x \# X"
  by (subst (asm) in_Inf_multiset_iff) auto

lemma set_mset_Sup:
  assumes "subset_mset.bdd_above A"
  shows   "set_mset (Sup A) = (\X\A. set_mset X)"
proof safe
  fix x assume "x \# Sup A"
  hence nonempty: "A \ {}" by (auto simp: Sup_multiset_empty)
  show "x \ (\X\A. set_mset X)"
  proof (rule ccontr)
    assume x: "x \ (\X\A. set_mset X)"
    have "count X x \ count (Sup A) x" if "X \ A" for X x
      using that by (intro mset_subset_eq_count subset_mset.cSup_upper assms)
    with x have "X \# Sup A - {#x#}" if "X \ A" for X
      using that by (auto simp: subseteq_mset_def algebra_simps not_in_iff)
    hence "Sup A \# Sup A - {#x#}" by (intro subset_mset.cSup_least nonempty)
    with \<open>x \<in># Sup A\<close> show False
      using mset_subset_diff_self by fastforce
  qed
next
  fix x X assume "x \ set_mset X" "X \ A"
  hence "{#x#} \# X" by auto
  also have "X \# Sup A" by (intro subset_mset.cSup_upper \X \ A\ assms)
  finally show "x \ set_mset (Sup A)" by simp
qed

lemma in_Sup_multiset_iff:
  assumes "subset_mset.bdd_above A"
  shows   "x \# Sup A \ (\X\A. x \# X)"
  by (simp add: assms set_mset_Sup)

lemma in_Sup_multisetD:
  assumes "x \# Sup A"
  shows   "\X\A. x \# X"
  using Sup_multiset_unbounded assms in_Sup_multiset_iff by fastforce

interpretation subset_mset: distrib_lattice "(\#)" "(\#)" "(\#)" "(\#)"
proof
  fix A B C :: "'a multiset"
  show "A \# (B \# C) = A \# B \# (A \# 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 \ bool) \ 'a multiset \ 'a multiset"
is "\P M. \x. if P x then M x else 0"
by (rule filter_preserves_multiset)

syntax (ASCII)
  "_MCollect" :: "pttrn \ 'a multiset \ bool \ 'a multiset"
    (\<open>(\<open>indent=1 notation=\<open>mixfix multiset comprehension\<close>\<close>{#_ :# _./ _#})\<close>)
syntax
  "_MCollect" :: "pttrn \ 'a multiset \ bool \ 'a multiset"
    (\<open>(\<open>indent=1 notation=\<open>mixfix multiset comprehension\<close>\<close>{#_ \<in># _./ _#})\<close>)
syntax_consts
  "_MCollect" == filter_mset
translations
  "{#x \# M. P#}" == "CONST filter_mset (\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 \ 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 \# N) = filter_mset P M \# filter_mset P N"
  by (rule multiset_eqI) simp

lemma filter_sup_mset[simp]: "filter_mset P (A \# B) = filter_mset P A \# 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 \# M"
  by (simp add: mset_subset_eqI)

lemma filter_mset_mono_strong:
  assumes "A \# B" "\x. x \# A \ P x \ Q x"
  shows   "filter_mset P A \# filter_mset Q B"
  by (rule mset_subset_eqI) (insert assms, auto simp: mset_subset_eq_count count_eq_zero_iff)

(* TODO: rename to filter_mset_mono_strong *)
lemma multiset_filter_mono:
  assumes "A \# B"
  shows "filter_mset f A \# filter_mset f B"
  using filter_mset_mono_strong[OF \<open>A \<subseteq># B\<close>] .

lemma filter_mset_eq_conv:
  "filter_mset P M = N \ N \# M \ (\b\#N. P b) \ (\a\#M - N. \ P a)" (is "?P \ ?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 \# 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_mset_eq_mempty_iff[simp]: "filter_mset P A = {#} \ (\x. x \# A \ \ P x)"
  by (auto simp: multiset_eq_iff count_eq_zero_iff)

lemma filter_filter_mset: "filter_mset P (filter_mset Q M) = {#x \# M. Q x \ P x#}"
  by (auto simp: multiset_eq_iff)

lemma
  filter_mset_True[simp]: "{#y \# M. True#} = M" and
  filter_mset_False[simp]: "{#y \# M. False#} = {#}"
  by (auto simp: multiset_eq_iff)

lemma filter_mset_cong0:
  assumes "\x. x \# M \ f x \ 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 \ count (filter_mset g M) x"
    using assms by (cases "x \# M") (simp_all add: not_in_iff)
next
  fix x
  show "count (filter_mset g M) x \ count (filter_mset f M) x"
    using assms by (cases "x \# M") (simp_all add: not_in_iff)
qed

lemma filter_mset_cong:
  assumes "M = M'" and "\x. x \# M' \ f x \ g x"
  shows "filter_mset f M = filter_mset g M'"
  unfolding \<open>M = M'\<close>
  using assms by (auto intro: filter_mset_cong0)

lemma filter_eq_replicate_mset: "{#y \# D. y = x#} = replicate_mset (count D x) x"
  by (induct D) (simp add: multiset_eqI)

subsubsection \<open>Size\<close>

definition wcount where "wcount f M = (\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 \ nat) \ 'a multiset \ 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 (\_. 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 \ sum (wcount f N) (A \ 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)
  by (metis add_implies_diff finite_set_mset inf.commute sum_wcount_Int)

lemma size_multiset_add_mset [simp]:
  "size_multiset f (add_mset a M) = Suc (f a) + size_multiset f M"
  by (metis add.commute add_mset_add_single size_multiset_single size_multiset_union)

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 \ 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 \ {#}) = (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 \ \a. a \# M"
  using all_not_in_conv by fastforce

lemma size_eq_Suc_imp_eq_union:
  assumes "size M = Suc n"
  shows "\a N. M = add_mset a N"
  by (metis assms insert_DiffM size_eq_Suc_imp_elem)

lemma size_mset_mono:
  fixes A B :: "'a multiset"
  assumes "A \# B"
  shows "size A \ 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) \ size M"
  by (rule size_mset_mono[OF multiset_filter_subset])

lemma size_Diff_submset:
  "M \# M' \ size (M' - M) = size M' - size(M::'a multiset)"
by (metis add_diff_cancel_left' size_union mset_subset_eq_exists_conv)

lemma size_lt_imp_ex_count_lt: "size M < size N \ \x \# N. count M x < count N x"
  by (metis count_eq_zero_iff leD not_le_imp_less not_less_zero size_mset_mono subseteq_mset_def)

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

theorem multiset_induct [case_names empty add, induct type: multiset]:
  assumes empty: "P {#}"
  assumes add: "\x M. P M \ 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: "\x M. P M \ (\y \# M. y \ x) \ 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: "\x M. P M \ (\y \# M. y \ x) \ 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 \ {#} \ \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 \# B \ B - {#c#} \ B"
  by (cases "B = {#}") (auto dest: multi_member_split)

lemma union_filter_mset_complement[simp]:
  "\x. P x = (\ Q x) \ filter_mset P M + filter_mset Q M = M"
  by (subst multiset_eq_iff) auto

lemma multiset_partition: "M = {#x \# M. P x#} + {#x \# M. \ P x#}"
  by simp

lemma mset_subset_size: "A \# B \ 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 \# B"
    by (meson add.prems subset_mset_def)
  then show ?case
    using add.prems subset_mset.less_eqE by fastforce
qed

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

lemma set_mset_subset_singletonD:
  assumes "set_mset A \ {x}"
  shows   "A = replicate_mset (size A) x"
  using assms by (induction A) auto

lemma count_conv_size_mset: "count A x = size (filter_mset (\y. y = x) A)"
  by (induction A) auto

lemma size_conv_count_bool_mset: "size A = count A True + count A False"
  by (induction A) 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 \# N}"
  using mset_subset_size wfp_def wfp_if_convertible_to_nat by blast

lemma wfp_subset_mset[simp]: "wfp (\#)"
  by (rule wf_subset_mset_rel[to_pred])

lemma full_multiset_induct [case_names less]:
  assumes ih: "\B. \(A::'a multiset). A \# B \ P A \ 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 \# A"
    and empty: "P {#}"
    and insert: "\a F. a \# A \ P F \ 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 \# A \ P F" and i: "add_mset x F \# A"
    show "P (add_mset x F)"
    proof (rule insert)
      from i show "x \# A" by (auto dest: mset_subset_eq_insertD)
      from i have "F \# A" by (auto dest: mset_subset_eq_insertD)
      with P show "P F" .
    qed
  qed
qed

subsection \<open>Least and greatest elements\<close>

context begin

qualified lemma
  assumes
    "M \ {#}" and
    "transp_on (set_mset M) R" and
    "totalp_on (set_mset M) R"
  shows
    bex_least_element: "(\l \# M. \x \# M. x \ l \ R l x)" and
    bex_greatest_element: "(\g \# M. \x \# M. x \ g \ R x g)"
  using assms
  by (auto intro: Finite_Set.bex_least_element Finite_Set.bex_greatest_element)

end

subsection \<open>The fold combinator\<close>

definition fold_mset :: "('a \ 'b \ 'b) \ 'b \ 'a multiset \ 'b"
where
  "fold_mset f s M = Finite_Set.fold (\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)

lemma fold_mset_single [simp]: "fold_mset f s {#x#} = f x 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 "\y. f y ^^ count M y"
    by (fact comp_fun_commute_funpow)
  interpret mset_union: comp_fun_commute "\y. f y ^^ count (add_mset x M) y"
    by (fact comp_fun_commute_funpow)
  show ?thesis
  proof (cases "x \ 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 (\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 \ N" "finite N" by auto
    then have "Finite_Set.fold (\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

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 *: "\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
--> --------------------

--> maximum size reached

--> --------------------

quality99%

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