# Theory Multiset

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

section ‹(Finite) multisets›

theory Multiset
imports Main
begin

subsection ‹The type of multisets›

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

typedef 'a multiset = "multiset :: ('a ⇒ nat) set"
morphisms count Abs_multiset
unfolding multiset_def
proof
show "(λx. 0::nat) ∈ {f. finite {x. f x > 0}}" by simp
qed

setup_lifting type_definition_multiset

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 ‹Preservation of the representing set @{term multiset}.›

lemma const0_in_multiset: "(λa. 0) ∈ multiset"

lemma only1_in_multiset: "(λb. if b = a then n else 0) ∈ multiset"

lemma union_preserves_multiset: "M ∈ multiset ⟹ N ∈ multiset ⟹ (λa. M a + N a) ∈ multiset"

lemma diff_preserves_multiset:
assumes "M ∈ multiset"
shows "(λa. M a - N a) ∈ multiset"
proof -
have "{x. N x < M x} ⊆ {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 ∈ multiset"
shows "(λx. if P x then M x else 0) ∈ multiset"
proof -
have "{x. (P x ⟶ 0 < M x) ∧ P x} ⊆ {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›

begin

lift_definition zero_multiset :: "'a multiset" is "λa. 0"
by (rule const0_in_multiset)

abbreviation Mempty :: "'a multiset" ("{#}") where
"Mempty ≡ 0"

lift_definition plus_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" is "λM N. (λa. M a + N a)"
by (rule union_preserves_multiset)

lift_definition minus_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" is "λ M N. λa. M a - N a"
by (rule diff_preserves_multiset)

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

end

context
begin

qualified definition is_empty :: "'a multiset ⇒ bool" where
[code_abbrev]: "is_empty A ⟷ A = {#}"

end

assumes M: ‹M ∈ multiset›
shows ‹(λb. if b = a then Suc (M b) else M b) ∈ multiset›
using assms by (simp add: multiset_def insert_Collect[symmetric])

lift_definition add_mset :: "'a ⇒ 'a multiset ⇒ 'a multiset" is
"λa M b. if b = a then Suc (M b) else M b"

syntax
"_multiset" :: "args ⇒ 'a multiset"    ("{#(_)#}")
translations
"{#x, xs#}" == "CONST add_mset x {#xs#}"
"{#x#}" == "CONST add_mset x {#}"

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

"count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"

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

lemma
by (auto simp: multiset_eq_iff)

by (auto simp: multiset_eq_iff)

by (auto simp: multiset_eq_iff)

subsection ‹Basic operations›

subsubsection ‹Conversion to set and membership›

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

abbreviation Melem :: "'a ⇒ 'a multiset ⇒ bool"
where "Melem a M ≡ a ∈ set_mset M"

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

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

abbreviation not_Melem :: "'a ⇒ 'a multiset ⇒ bool"
where "not_Melem a M ≡ a ∉ set_mset M"

notation
not_Melem  ("op ∉#") and
not_Melem  ("(_/ ∉# _)" [51, 51] 50)

notation  (ASCII)
not_Melem  ("op ~:#") and
not_Melem  ("(_/ ~:# _)" [51, 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"      ("(3∀_∈#_./ _)" [0, 0, 10] 10)
"_MBex"        :: "pttrn ⇒ 'a set ⇒ bool ⇒ bool"      ("(3∃_∈#_./ _)" [0, 0, 10] 10)

syntax  (ASCII)
"_MBall"       :: "pttrn ⇒ 'a set ⇒ bool ⇒ bool"      ("(3∀_:#_./ _)" [0, 0, 10] 10)
"_MBex"        :: "pttrn ⇒ 'a set ⇒ bool ⇒ bool"      ("(3∃_:#_./ _)" [0, 0, 10] 10)

translations
"∀x∈#A. P" ⇌ "CONST Multiset.Ball A (λx. P)"
"∃x∈#A. P" ⇌ "CONST Multiset.Bex A (λx. P)"

lemma count_eq_zero_iff:
"count M x = 0 ⟷ x ∉# M"

lemma not_in_iff:
"x ∉# M ⟷ count M x = 0"

lemma count_greater_zero_iff [simp]:
"count M x > 0 ⟷ x ∈# M"

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"

lemma count_greater_eq_one_iff [simp]:
"count M x ≥ 1 ⟷ x ∈# M"
by simp

lemma set_mset_empty [simp]:
"set_mset {#} = {}"

lemma set_mset_single:
"set_mset {#b#} = {b}"

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 add: multiset_def)

lemma set_mset_add_mset_insert [simp]: ‹set_mset (add_mset a A) = insert a (set_mset A)›
by (auto simp del: count_greater_eq_Suc_zero_iff
simp: count_greater_eq_Suc_zero_iff[symmetric] 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

subsubsection ‹Union›

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

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

by (auto simp: multiset_eq_iff)

by (auto simp: multiset_eq_iff)

subsubsection ‹Difference›

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 (auto simp: multiset_eq_iff)

lemma in_diff_count:
"a ∈# M - N ⟷ count N a < count M a"

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"
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"
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"
finally show ?thesis by simp
qed

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

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)"

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

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

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

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#}"

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)"

lemma diff_union_single_conv:
"a ∈# J ⟹ I + J - {#a#} = I + (J - {#a#})"

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

subsubsection ‹Equality of multisets›

lemma single_eq_single [simp]: "{#a#} = {#b#} ⟷ a = b"

lemma union_eq_empty [iff]: "M + N = {#} ⟷ M = {#} ∧ N = {#}"

lemma empty_eq_union [iff]: "{#} = M + N ⟷ M = {#} ∧ N = {#}"

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#}"

lemma union_single_eq_member: "add_mset x M = N ⟹ x ∈# N"
by auto

"add_mset a (N - {#a#}) = (if a ∈# N then N else add_mset a N)"

lemma add_mset_remove_trivial_eq: ‹N = add_mset a (N - {#a#}) ⟷ a ∈# N›

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
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)

"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
show ?rhs if ?lhs
proof (cases "a = b")
case True with ‹?lhs› show ?thesis by simp
next
case False
from ‹?lhs› have "a ∈# add_mset b N" by (rule union_single_eq_member)
with False have "a ∈# N" by auto
moreover from ‹?lhs› 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 = {#}"

lemma single_eq_add_mset [iff]: "{#a#} = add_mset b M ⟷ b = a ∧ M = {#}"

lemma insert_noteq_member:
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

(M = N ∧ a = b ∨ (∃K. M = add_mset b K ∧ N = add_mset a K))"

lemma multi_member_split: "x ∈# M ⟹ ∃A. M = add_mset x A"
by (rule exI [where x = "M - {#x#}"]) simp

assumes "c ∈# B"
and "b ≠ c"
shows "add_mset b (B - {#c#}) = add_mset b B - {#c#}"
proof -
from ‹c ∈# B› obtain A where B: "B = add_mset c A"
by (blast dest: multi_member_split)
by (simp add: ‹b ≠ c›)
then show ?thesis using B by simp
qed

"add_mset x M = {#y#} ⟷ M = {#} ∧ x = y"
by auto

subsubsection ‹Pointwise ordering induced by count›

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 "≤#" 50) and
supseteq_mset  (infix "≥#" 50)

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

interpretation subset_mset: ordered_ab_semigroup_add_imp_le "op +" "op -" "op ⊆#" "op ⊂#"
by standard (auto simp add: subset_mset_def subseteq_mset_def multiset_eq_iff intro: order_trans antisym)

interpretation subset_mset: ordered_ab_semigroup_monoid_add_imp_le "op +" 0 "op -" "op ≤#" "op <#"
by standard

lemma mset_subset_eqI:
"(⋀a. count A a ≤ count B a) ⟹ A ⊆# B"

lemma mset_subset_eq_count:
"A ⊆# B ⟹ count A a ≤ count B a"

lemma mset_subset_eq_exists_conv: "(A::'a multiset) ⊆# B ⟷ (∃C. B = A + C)"
unfolding subseteq_mset_def
apply (rule iffI)
apply (rule exI [where x = "B - A"])
apply (auto intro: multiset_eq_iff [THEN iffD2])
done

interpretation subset_mset: ordered_cancel_comm_monoid_diff "op +" 0 "op ≤#" "op <#" "op -"
by standard (simp, fact mset_subset_eq_exists_conv)

lemma mset_subset_eq_mono_add_right_cancel: "(A::'a multiset) + C ⊆# B + C ⟷ A ⊆# B"

lemma mset_subset_eq_mono_add_left_cancel: "C + (A::'a multiset) ⊆# C + B ⟷ A ⊆# B"

lemma mset_subset_eq_mono_add: "(A::'a multiset) ⊆# B ⟹ C ⊆# D ⟹ A + C ⊆# B + D"

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 multiset_diff_union_assoc:
fixes A B C D :: "'a multiset"
shows "C ⊆# B ⟹ A + B - C = A + (B - C)"

lemma mset_subset_eq_multiset_union_diff_commute:
fixes A B C D :: "'a multiset"
shows "B ⊆# A ⟹ A - B + C = A + C - B"

lemma diff_subset_eq_self[simp]:
"(M::'a multiset) - N ⊆# M"

lemma mset_subset_eqD:
assumes "A ⊆# B" and "x ∈# A"
shows "x ∈# B"
proof -
from ‹x ∈# A› have "count A x > 0" by simp
also from ‹A ⊆# B› have "count A x ≤ count B x"
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:
"add_mset x A ⊆# B ⟹ x ∈# B ∧ A ⊂# B"
apply (rule conjI)
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
apply safe
apply (erule_tac x = a in allE)
apply (auto split: if_split_asm)
done

lemma mset_subset_insertD:
"add_mset x A ⊂# 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)

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 le_diff_conv2 [of "Suc 0" "count B a" "count A a"]
apply (auto simp add: subseteq_mset_def not_in_iff Suc_le_eq)
apply (rule ccontr)
done

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 (auto simp: subset_mset_def subseteq_mset_def)

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

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 ‹a = b›
by (auto dest: mset_subset_eq_insertD)
then show "M={#} ∧ a=b"
qed simp

subsubsection ‹Intersection and bounded union›

definition inf_subset_mset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" (infixl "∩#" 70) where
multiset_inter_def: "inf_subset_mset A B = A - (A - B)"

interpretation subset_mset: semilattice_inf inf_subset_mset "op ⊆#" "op ⊂#"
proof -
have [simp]: "m ≤ n ⟹ m ≤ q ⟹ m ≤ n - (n - q)" for m n q :: nat
by arith
show "class.semilattice_inf op ∩# op ⊆# op ⊂#"
by standard (auto simp add: multiset_inter_def subseteq_mset_def)
qed
― ‹FIXME: avoid junk stemming from type class interpretation›

definition sup_subset_mset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset"(infixl "∪#" 70)
where "sup_subset_mset A B = A + (B - A)" ― ‹FIXME irregular fact name›

interpretation subset_mset: semilattice_sup sup_subset_mset "op ⊆#" "op ⊂#"
proof -
have [simp]: "m ≤ n ⟹ q ≤ n ⟹ m + (q - m) ≤ n" for m n q :: nat
by arith
show "class.semilattice_sup op ∪# op ⊆# op ⊂#"
by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
qed
― ‹FIXME: avoid junk stemming from type class interpretation›

interpretation subset_mset: bounded_lattice_bot "op ∩#" "op ⊆#" "op ⊂#"
"op ∪#" "{#}"
by standard auto

lemma multiset_inter_count [simp]:
fixes A B :: "'a multiset"
shows "count (A ∩# B) x = min (count A x) (count B x)"

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

lemma diff_intersect_left_idem [simp]:
"M - M ∩# N = M - N"

lemma diff_intersect_right_idem [simp]:
"M - N ∩# M = M - N"

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"
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)" (is "?P ⟷ ?Q")
proof
assume ?P
show ?Q
proof
fix a
from ‹?P› have "min (count A a) (count B a) = 0"
then have "count A a = 0 ∨ count B a = 0"
by (cases "count A a ≤ count B a") (simp_all add: min_def)
then show "a ∉# A ∨ a ∉# B"
qed
next
assume ?Q
show ?P
proof (rule multiset_eqI)
fix a
from ‹?Q› have "count A a = 0 ∨ count B a = 0"
then show "count (A ∩# B) a = count {#} a"
by auto
qed
qed

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)

"add_mset a A ∩# B = {#} ⟷ a ∉# B ∧ A ∩# B = {#}"
"{#} = add_mset a A ∩# B ⟷ a ∉# B ∧ {#} = A ∩# B"
by (auto simp: disjunct_not_in)

"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"

lemma inter_add_left2: "x ∈# N ⟹ (add_mset x M) ∩# N = add_mset x (M ∩# (N - {#x#}))"

lemma inter_add_right1: "¬ x ∈# N ⟹ N ∩# (add_mset x M) = N ∩# M"

lemma inter_add_right2: "x ∈# N ⟹ N ∩# (add_mset x M) = add_mset x ((N - {#x#}) ∩# M)"

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"
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)"

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"

lemma sup_subset_mset_count [simp]: ― ‹FIXME irregular fact name›
"count (A ∪# B) x = max (count A x) (count B x)"

lemma set_mset_sup [simp]:
"set_mset (A ∪# B) = set_mset A ∪ set_mset B"
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] sup_subset_mset_count)

lemma sup_union_left1 [simp]: "¬ x ∈# N ⟹ (add_mset x M) ∪# N = add_mset x (M ∪# N)"

lemma sup_union_left2: "x ∈# N ⟹ (add_mset x M) ∪# N = add_mset x (M ∪# (N - {#x#}))"

lemma sup_union_right1 [simp]: "¬ x ∈# N ⟹ N ∪# (add_mset x M) = add_mset x (N ∪# M)"

lemma sup_union_right2: "x ∈# N ⟹ N ∪# (add_mset x M) = add_mset x ((N - {#x#}) ∪# M)"

lemma sup_union_distrib_left:
"A ∪# B + C = (A + C) ∪# (B + C)"

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"

lemma union_diff_sup_eq_inter:
"A + B - A ∪# B = A ∩# B"

by (auto simp: multiset_eq_iff max_def)

subsubsection ‹Subset is an order›

interpretation subset_mset: order "op ⊆#" "op ⊂#" by unfold_locales

subsection ‹Replicate and repeat operations›

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

fun repeat_mset :: "nat ⇒ 'a multiset ⇒ 'a multiset" where
"repeat_mset 0 _ = {#}" |
"repeat_mset (Suc n) A = A + repeat_mset n A"

lemma count_repeat_mset [simp]: "count (repeat_mset i A) a = i * count A a"
by (induction i) auto

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')

"repeat_mset i u + (repeat_mset j u + k) = repeat_mset (i+j) u + k"

lemma repeat_mset_distrib:
"repeat_mset (m + n) A = repeat_mset m A + repeat_mset n A"

lemma repeat_mset_distrib2[simp]:
"repeat_mset n (A + B) = repeat_mset n A + repeat_mset n B"

lemma repeat_mset_replicate_mset[simp]:
"repeat_mset n {#a#} = replicate_mset n a"
by (auto simp: multiset_eq_iff)

"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 (induction n) simp_all

subsubsection ‹Simprocs›

"j ≤ (i::nat) ⟹ ((repeat_mset i u + m) - (repeat_mset j u + n)) = ((repeat_mset (i-j) u + m) - n)"

"i ≤ (j::nat) ⟹ ((repeat_mset i u + m) - (repeat_mset j u + n)) = (m - (repeat_mset (j-i) u + n))"

"j ≤ (i::nat) ⟹ (repeat_mset i u + m = repeat_mset j u + n) = (repeat_mset (i-j) u + m = n)"

"i ≤ (j::nat) ⟹ (repeat_mset i u + m = repeat_mset j u + n) = (m = repeat_mset (j-i) u + n)"

"j ≤ (i::nat) ⟹ (repeat_mset i u + m ⊆# repeat_mset j u + n) = (repeat_mset (i-j) u + m ⊆# n)"

"i ≤ (j::nat) ⟹ (repeat_mset i u + m ⊆# repeat_mset j u + n) = (m ⊆# repeat_mset (j-i) u + n)"

"j ≤ (i::nat) ⟹ (repeat_mset i u + m ⊂# repeat_mset j u + n) = (repeat_mset (i-j) u + m ⊂# n)"

"i ≤ (j::nat) ⟹ (repeat_mset i u + m ⊂# repeat_mset j u + n) = (m ⊂# repeat_mset (j-i) u + n)"

ML_file "multiset_simprocs_util.ML"
ML_file "multiset_simprocs.ML"

simproc_setup mseteq_cancel_numerals
("(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") =
‹fn phi => Multiset_Simprocs.eq_cancel_msets›

simproc_setup msetless_cancel_numerals
("(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") =
‹fn phi => Multiset_Simprocs.subset_cancel_msets›

simproc_setup msetle_cancel_numerals
("(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") =
‹fn phi => Multiset_Simprocs.subseteq_cancel_msets›

simproc_setup msetdiff_cancel_numerals
("((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") =
‹fn phi => Multiset_Simprocs.diff_cancel_msets›

subsubsection ‹Conditionally complete lattice›

instantiation multiset :: (type) Inf
begin

lift_definition Inf_multiset :: "'a multiset set ⇒ 'a multiset" is
"λA i. if A = {} then 0 else Inf ((λf. f i) ` A)"
proof -
fix A :: "('a ⇒ nat) set" assume *: "⋀x. x ∈ A ⟹ x ∈ multiset"
have "finite {i. (if A = {} then 0 else Inf ((λf. f i) ` A)) > 0}" unfolding multiset_def
proof (cases "A = {}")
case False
then obtain f where "f ∈ A" by blast
hence "{i. Inf ((λf. f i) ` A) > 0} ⊆ {i. f i > 0}"
by (auto intro: less_le_trans[OF _ cInf_lower])
moreover from ‹f ∈ A› * have "finite …" by (simp add: multiset_def)
ultimately have "finite {i. Inf ((λf. f i) ` A) > 0}" by (rule finite_subset)
with False show ?thesis by simp
qed simp_all
thus "(λi. if A = {} then 0 else INF f:A. f i) ∈ multiset" by (simp add: multiset_def)
qed

instance ..

end

lemma Inf_multiset_empty: "Inf {} = {#}"
by transfer simp_all

lemma count_Inf_multiset_nonempty: "A ≠ {} ⟹ count (Inf A) x = Inf ((λX. 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 (λi. Sup ((λX. count X i) ` A)) else {#})"

lemma Sup_multiset_empty: "Sup {} = {#}"

lemma Sup_multiset_unbounded: "¬subset_mset.bdd_above A ⟹ Sup A = {#}"

instance ..

end

lemma bdd_above_multiset_imp_bdd_above_count:
assumes "subset_mset.bdd_above (A :: 'a multiset set)"
shows   "bdd_above ((λX. count X x) ` A)"
proof -
from assms obtain Y where Y: "∀X∈A. X ⊆# Y"
by (auto simp: subset_mset.bdd_above_def)
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 (auto simp: subset_mset.bdd_above_def)
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:
assumes "A ≠ {}" "subset_mset.bdd_above A"
shows   "(λi. SUP X:A. count X i) ∈ multiset"
unfolding multiset_def
proof
have "{i. Sup ((λX. 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 assms 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 assms have "finite …" by (rule bdd_above_multiset_imp_finite_support)
ultimately show "finite {i. Sup ((λX. count X i) ` A) > 0}" by (rule finite_subset)
qed

lemma count_Sup_multiset_nonempty:
assumes "A ≠ {}" "subset_mset.bdd_above A"
shows   "count (Sup A) x = (SUP X:A. count X x)"
using assms by (simp add: Sup_multiset_def Abs_multiset_inverse Sup_multiset_in_multiset)

interpretation subset_mset: conditionally_complete_lattice Inf Sup "op ∩#" "op ⊆#" "op ⊂#" "op ∪#"
proof
fix X :: "'a multiset" and A
assume "X ∈ A"
show "Inf A ⊆# X"
proof (rule mset_subset_eqI)
fix x
from ‹X ∈ A› have "A ≠ {}" by auto
hence "count (Inf A) x = (INF X:A. count X x)"
also from ‹X ∈ A› have "… ≤ count X x"
by (intro cInf_lower) simp_all
finally show "count (Inf A) x ≤ count X x" .
qed
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 ‹A ≠ {}› 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 (rule subset_mset.bdd_aboveI[of _ X])
show "Sup A ⊆# X"
proof (rule mset_subset_eqI)
fix x
from count_Sup_multiset_nonempty[OF ‹A ≠ {}› 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

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 ‹x ∈# Inf A› have "{#x#} ⊆# Inf A" by auto
also from ‹X ∈ A› have "… ⊆# 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 ‹x ∈# Sup A› show False
by (auto simp: subseteq_mset_def count_greater_zero_iff [symmetric]
simp del: count_greater_zero_iff dest!: spec[of _ x])
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)"
proof -
from assms have "set_mset (Sup A) = (⋃X∈A. set_mset X)" by (rule set_mset_Sup)
also have "x ∈ … ⟷ (∃X∈A. x ∈# X)" by simp
finally show ?thesis .
qed

lemma in_Sup_multisetD:
assumes "x ∈# Sup A"
shows   "∃X∈A. x ∈# X"
proof -
have "subset_mset.bdd_above A"
by (rule ccontr) (insert assms, simp_all add: Sup_multiset_unbounded)
with assms show ?thesis by (simp add: in_Sup_multiset_iff)
qed

interpretation subset_mset: distrib_lattice "op ∩#" "op ⊆#" "op ⊂#" "op ∪#"
proof
fix A B C :: "'a multiset"
show "A ∪# (B ∩# C) = A ∪# B ∩# (A ∪# C)"
by (intro multiset_eqI) simp_all
qed

subsubsection ‹Filter (with comprehension syntax)›

text ‹Multiset comprehension›

lift_definition filter_mset :: "('a ⇒ bool) ⇒ 'a multiset ⇒ 'a multiset"
is "λP M. λx. if P x then M x else 0"
by (rule filter_preserves_multiset)

syntax (ASCII)
"_MCollect" :: "pttrn ⇒ 'a multiset ⇒ bool ⇒ 'a multiset"    ("(1{#_ :# _./ _#})")
syntax
"_MCollect" :: "pttrn ⇒ 'a multiset ⇒ bool ⇒ 'a multiset"    ("(1{#_ ∈# _./ _#})")
translations
"{#x ∈# M. P#}" == "CONST filter_mset (λx. P) M"

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

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

"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"

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

lemma filter_mset_eq_conv:
"filter_mset P M = N ⟷ 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"
then have MN: "M - N = Q" by simp
show ?P
proof (rule multiset_eqI)
fix a
from ‹?Q› MN have *: "¬ P a ⟹ a ∉# N" "P a ⟹ a ∉# Q"
by auto
show "count (filter_mset P M) a = count N a"
proof (cases "a ∈# M")
case True
with * show ?thesis
next
case False then have "count M a = 0"
with M show ?thesis by simp
qed
qed
qed

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)

subsubsection ‹Size›

definition wcount where "wcount f M = (λx. count M x * Suc (f x))"

lemma wcount_union: "wcount f (M + N) a = wcount f M a + wcount f N a"

"wcount f (add_mset x M) a = (if x = a then Suc (f a) else 0) + wcount f M a"

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 (λ_. 0)"
instance ..

end

lemma size_multiset_empty [simp]: "size_multiset f {#} = 0"

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

lemma size_multiset_single : "size_multiset f {#b#} = Suc (f b)"

lemma size_single: "size {#b#} = 1"

lemma sum_wcount_Int:
"finite A ⟹ sum (wcount f N) (A ∩ set_mset N) = sum (wcount f N) A"
by (induct rule: finite_induct)

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

"size_multiset f (add_mset a M) = Suc (f a) + size_multiset f M"

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

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 = {#})"

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"
apply (drule sum_SucD)
apply auto
done

lemma size_eq_Suc_imp_eq_union:
assumes "size M = Suc n"
shows "∃a N. M = add_mset a N"
proof -
from assms obtain a where "a ∈# M"
by (erule size_eq_Suc_imp_elem [THEN exE])
then have "M = add_mset a (M - {#a#})" by simp
then show ?thesis by blast
qed

lemma size_mset_mono:
fixes A B :: "'a multiset"
assumes "A ⊆# 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)"

subsection ‹Induction and case splits›

theorem multiset_induct [case_names empty add, induct type: multiset]:
assumes empty: "P {#}"
shows "P M"
proof (induct n ≡ "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 ‹Suc k = size M› [symmetric]
using size_eq_Suc_imp_eq_union by fast
with Suc add show "P M" by simp
qed

lemma multi_nonempty_split: "M ≠ {#} ⟹ ∃A a. M = add_mset a A"
by (induct M) auto

lemma multiset_cases [cases type]:
obtains (empty) "M = {#}"
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 multiset_partition: "M = {# x∈#M. P x #} + {# x∈#M. ¬ P x #}"
apply (subst multiset_eq_iff)
apply auto
done

lemma mset_subset_size: "(A::'a multiset) ⊂# B ⟹ size A < size B"
proof (induct A arbitrary: B)
case (empty M)
then have "M ≠ {#}" by (simp add: subset_mset.zero_less_iff_neq_zero)
then obtain M' x where "M = add_mset x M'"
by (blast dest: multi_nonempty_split)
then show ?case by simp
next
have IH: "⋀B. S ⊂# B ⟹ size S < size B" by fact
have SxsubT: "add_mset x S ⊂# T" by fact
then have "x ∈# T" and "S ⊂# T"
by (auto dest: mset_subset_insertD)
then obtain T' where T: "T = add_mset x T'"
by (blast dest: multi_member_split)
then have "S ⊂# T'" using SxsubT
by simp
then have "size S < size T'" using IH by simp
then show ?case using T by simp
qed

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

subsubsection ‹Strong induction and subset induction for multisets›

text ‹Well-foundedness of strict subset relation›

lemma wf_subset_mset_rel: "wf {(M, N :: 'a multiset). M ⊂# N}"
apply (rule wf_measure [THEN wf_subset, where f1=size])
apply (clarsimp simp: measure_def inv_image_def mset_subset_size)
done

lemma full_multiset_induct [case_names less]:
assumes ih: "⋀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 ‹F ⊆# A›
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"
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 ‹The fold combinator›

definition fold_mset :: "('a ⇒ 'b ⇒ 'b) ⇒ 'b ⇒ 'a multiset ⇒ 'b"
where
"fold_mset f s M = Finite_Set.fold (λx. f x ^^ count M x) s (set_mset M)"

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

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 "λy. f y ^^ count M y"
by (fact comp_fun_commute_funpow)
interpret mset_union: comp_fun_commute "λy. 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"
from False have "Finite_Set.fold (λy. f y ^^ count (add_mset x M) y) s (set_mset M) =
Finite_Set.fold (λy. f y ^^ count M y) s (set_mset M)"
by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
with False * show ?thesis
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 (λy. f y ^^ count (add_mset x M) y) s N =
Finite_Set.fold (λy. f y ^^ count M y) s N"
by (auto intro!: Finite_Set.fold_cong comp_fun_commute_funpow)
qed
qed

corollary fold_mset_single: "fold_mset f s {#x#} = f x s"
by simp

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

lemma fold_mset_union [simp]: "fold_mset f s (M + N) = fold_mset f (fold_mset f s M) N"
by (induct M) (simp_all add: fold_mset_fun_left_comm)

lemma fold_mset_fusion:
assumes "comp_fun_commute g"
and *: "⋀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

proof -
by standard auto
show ?thesis
by (induction B) auto
qed

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 ‹left_commutative› with ‹F›, the
would be code thms for @{const fold_mset} become thms like
@{term "fold_mset F z {#} = z"} where ‹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 ‹F›. See the image operator below.
›

subsection ‹Image›

definition image_mset :: "('a ⇒ 'b) ⇒ 'a multiset ⇒ 'b multiset" where
"image_mset f = fold_mset (add_mset ∘ f) {#}"

lemma comp_fun_commute_mset_image: "comp_fun_commute (add_mset ∘ f)"
proof

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

lemma image_mset_single: "image_mset f {#x#} = {#f x#}"
proof -
by (fact comp_fun_commute_mset_image)
show ?thesis by (simp add: image_mset_def)
qed

lemma image_mset_union [simp]: "image_mset f (M + N) = image_mset f M + image_mset f N"
proof -
by (fact comp_fun_commute_mset_image)
show ?thesis by (induct N) (simp_all add: image_mset_def)
qed

"image_mset f (add_mset a M) = add_mset (f a) (image_mset f M)"

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

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

lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} ⟷ M = {#}"
by (cases M) auto

lemma image_mset_If:
"image_mset (λx. if P x then f x else g x) A =
image_mset f (filter_mset P A) + image_mset g (filter_mset (λx. ¬P x) A)"
by (induction A) auto

lemma image_mset_Diff:
assumes "B ⊆# A"
shows   "image_mset f (A - B) = image_mset f A - image_mset f B"
proof -
have "image_mset f (A - B + B) = image_mset f (A - B) + image_mset f B"
by simp
also from assms have "A - B + B = A"
finally show ?thesis by simp
qed

lemma count_image_mset:
"count (image_mset f A) x = (∑y∈f -` {x} ∩ set_mset A. count A y)"
proof (induction A)
case empty
then show ?case by simp
next
moreover have *: "(if x = y then Suc n else n) = n + (if x = y then 1 else 0)" for n y
by simp
ultimately show ?case
by (auto simp: sum.distrib sum.delta' intro!: sum.mono_neutral_left)
qed

lemma image_mset_subseteq_mono: "A ⊆# B ⟹ image_mset f A ⊆# image_mset f B"

syntax (ASCII)
"_comprehension_mset" :: "'a ⇒ 'b ⇒ 'b multiset ⇒ 'a multiset"  ("({#_/. _ :# _#})")
syntax
"_comprehension_mset" :: "'a ⇒ 'b ⇒ 'b multiset ⇒ 'a multiset"  ("({#_/. _ ∈# _#})")
translations
"{#e. x ∈# M#}" ⇌ "CONST image_mset (λx. e) M"

syntax (ASCII)
"_comprehension_mset'" :: "'a ⇒ 'b ⇒ 'b multiset ⇒ bool ⇒ 'a multiset"  ("({#_/ | _ :# _./ _#})")
syntax
"_comprehension_mset'" :: "'a ⇒ 'b ⇒ 'b multiset ⇒ bool ⇒ '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#}"}.
›

lemma in_image_mset: "y ∈# {#f x. x ∈# M#} ⟷ y ∈ f ` set_mset M"
by (metis set_image_mset)

functor image_mset: image_mset
proof -
fix f g show "image_mset f ∘ image_mset g = image_mset (f ∘ g)"
proof
fix A
show "(image_mset f ∘ image_mset g) A = image_mset (f ∘ g) A"
by (induct A) simp_all
qed
show "image_mset id = id"
proof
fix A
show "image_mset id A = id A"
by (induct A) simp_all
qed
qed

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

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

lemma image_mset_cong: "(⋀x. x ∈# M ⟹ f x = g x) ⟹ {#f x. x ∈# M#} = {#g x. x ∈# M#}"
by (induct M) auto

lemma image_mset_cong_pair:
"(∀x y. (x, y) ∈# M ⟶ f x y = g x y) ⟹ {#f x y. (x, y) ∈# M#} = {#g x y. (x, y) ∈# M#}"
by (metis image_mset_cong split_cong)

subsection ‹Further conversions›

primrec mset :: "'a list ⇒ 'a multiset" where
"mset [] = {#}" |
"mset (a # x) = add_mset a (mset x)"

lemma in_multiset_in_set:
"x ∈# mset xs ⟷ x ∈ set xs"
by (induct xs) simp_all

lemma count_mset:
"count (mset xs) x = length (filter (λy. x = y) xs)"
by (induct xs) simp_all

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

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

lemma mset_single_iff[iff]: "mset xs = {#x#} ⟷ xs = [x]"
by (cases xs) auto

lemma mset_single_iff_right[iff]: "{#x#} = mset xs ⟷ xs = [x]"
by (cases xs) auto

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

lemma set_mset_comp_mset [simp]: "set_mset ∘ mset = set"

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

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

lemma mset_filter: "mset (filter P xs) = {#x ∈# mset xs. P x #}"
by (induct xs) simp_all

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

lemma surj_mset: "surj mset"
apply (unfold surj_def)
apply (rule allI)
apply (rule_tac M = y in multiset_induct)
apply auto
apply (rule_tac x = "x # xa" in exI)
apply auto
done

lemma distinct_count_atmost_1:
"distinct x = (∀a. count (mset x) a = (if a ∈ set x then 1 else 0))"
proof (induct x)
case Nil then show ?case by simp
next
case (Cons x xs) show ?case (is "?lhs ⟷ ?rhs")
proof
assume ?lhs then show ?rhs using Cons by simp
next
assume ?rhs then have "x ∉ set xs"
by (simp split: if_splits)
moreover from ‹?rhs› have "(∀a. count (mset xs) a =
(if a ∈ set xs then 1 else 0))"
by (auto split: if_splits simp add: count_eq_zero_iff)
ultimately show ?lhs using Cons by simp
qed
qed

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

lemma set_eq_iff_mset_eq_distinct:
"distinct x ⟹ distinct y ⟹
(set x = set y) = (mset x = mset y)"
by (auto simp: multiset_eq_iff distinct_count_atmost_1)

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

lemma mset_compl_union [simp]: "mset [x←xs. P x] + mset [x←xs. ¬P x] = mset xs"
by (induct xs) auto

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

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

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

lemma mset_eq_length_filter:
assumes "mset xs = mset ys"
shows "length (filter (λx. z = x) xs) = length (filter (λy. z = y) ys)"
using assms by (metis count_mset)

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

lemma mset_insort [simp]: "mset (insort x xs) = add_mset x (mset xs)"
by (induct xs) simp_all

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

defines mset_set = "folding.F add_mset {#}"

lemma count_mset_set [simp]:
"finite A ⟹ x ∈ A ⟹ count (mset_set A) x = 1" (is "PROP ?P")
"¬ finite A ⟹ count (mset_set A) x = 0" (is "PROP ?Q")
"x ∉ A ⟹ count (mset_set A) x = 0" (is "PROP ?R")
proof -
have *: "count (mset_set A) x = 0" if "x ∉ A" for A
proof (cases "finite A")
case False then show ?thesis by simp
next
case True from True ‹x ∉ A› show ?thesis by (induct A) auto
qed
then show "PROP ?P" "PROP ?Q" "PROP ?R"
by (auto elim!: Set.set_insert)
qed ― ‹TODO: maybe define @{const mset_set} also in terms of @{const Abs_multiset}›

lemma elem_mset_set[simp, intro]: "finite A ⟹ x ∈# mset_set A ⟷ x ∈ A"
by (induct A rule: finite_induct) simp_all

lemma mset_set_Union:
"finite A ⟹ finite B ⟹ A ∩ B = {} ⟹ mset_set (A ∪ B) = mset_set A + mset_set B"
by (induction A rule: finite_induct) auto

lemma filter_mset_mset_set [simp]:
"finite A ⟹ filter_mset P (mset_set A) = mset_set {x∈A. P x}"
proof (induction A rule: finite_induct)
case (insert x A)
from insert.hyps have "filter_mset P (mset_set (insert x A)) =
filter_mset P (mset_set A) + mset_set (if P x then {x} else {})"
by simp
also have "filter_mset P (mset_set A) = mset_set {x∈A. P x}"
by (rule insert.IH)
also from insert.hyps
have "… + mset_set (if P x then {x} else {}) =
mset_set ({x ∈ A. P x} ∪ (if P x then {x} else {}))" (is "_ = mset_set ?A")
by (intro mset_set_Union [symmetric]) simp_all
also from insert.hyps have "?A = {y∈insert x A. P y}" by auto
finally show ?case .
qed simp_all

lemma mset_set_Diff:
assumes "finite A" "B ⊆ A"
shows  "mset_set (A - B) = mset_set A - mset_set B"
proof -
from assms have "mset_set ((A - B) ∪ B) = mset_set (A - B) + mset_set B"
by (intro mset_set_Union) (auto dest: finite_subset)
also from assms have "A - B ∪ B = A" by blast
finally show ?thesis by simp
qed

lemma mset_set_set: "distinct xs ⟹ mset_set (set xs) = mset xs"
by (induction xs) simp_all

context linorder
begin

definition sorted_list_of_multiset :: "'a multiset ⇒ 'a list"
where
"sorted_list_of_multiset M = fold_mset insort [] M"

lemma sorted_list_of_multiset_empty [simp]:
"sorted_list_of_multiset {#} = []"

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

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

end

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

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

lemma finite_set_mset_mset_set[simp]:
"finite A ⟹ set_mset (mset_set A) = A"
by (induct A rule: finite_induct) simp_all

lemma mset_set_empty_iff: "mset_set A = {#} ⟷ A = {} ∨ infinite A"
using finite_set_mset_mset_set by fastforce

lemma infinite_set_mset_mset_set:
"¬ finite A ⟹ set_mset (mset_set A) = {}"
by simp

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

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

lemma mset_upt [simp]: "mset [m..<n] = mset_set {m..<n}"
by (induction n) (simp_all add: atLeastLessThanSuc)

lemma image_mset_map_of:
"distinct (map fst xs) ⟹ {#the (map_of xs i). i ∈# mset (map fst xs)#} = mset (map snd xs)"
proof (induction xs)
case (Cons x xs)
have "{#the (map_of (x # xs) i). i ∈# mset (map fst (x # xs))#} =
add_mset (snd x) {#the (if i = fst x then Some (snd x) else map_of xs i).
i ∈# mset (map fst xs)#}" (is "_ = add_mset _ ?A") by simp
also from Cons.prems have "?A = {#the (map_of xs i). i :# mset (map fst xs)#}"
by (cases x, intro image_mset_cong) (auto simp: in_multiset_in_set)
also from Cons.prems have "… = mset (map snd xs)" by (intro Cons.IH) simp_all
finally show ?case by simp
qed simp_all

(* Contributed by Lukas Bulwahn *)
lemma image_mset_mset_set:
assumes "inj_on f A"
shows "image_mset f (mset_set A) = mset_set (f ` A)"
proof cases
assume "finite A"
from this ‹inj_on f A› show ?thesis
by (induct A) auto
next
assume "infinite A"
from this ‹inj_on f A› have "infinite (f ` A)"
using finite_imageD by blast
from ‹infinite A› ‹infinite (f ` A)› show ?thesis by simp
qed

subsection ‹More properties of the replicate and repeat operations›

lemma in_replicate_mset[simp]: "x ∈# replicate_mset n y ⟷ n > 0 ∧ x = y"
unfolding replicate_mset_def by (induct n) auto

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

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

lemma count_le_replicate_mset_subset_eq: "n ≤ count M x ⟷ replicate_mset n x ⊆# M"
by (auto simp add: mset_subset_eqI) (metis count_replicate_mset subseteq_mset_def)

lemma filter_eq_replicate_mset: "{#y ∈# D. y = x#} = replicate_mset (count D x) x"
by (induct D) simp_all

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

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

lemma replicate_mset_eq_iff:
"replicate_mset m a = replicate_mset n b ⟷
m = 0 ∧ n = 0 ∨ m = n ∧ a = b"

lemma repeat_mset_cancel1: "repeat_mset a A = repeat_mset a B ⟷ A = B ∨ a = 0"
by (auto simp: multiset_eq_iff)

lemma repeat_mset_cancel2: "repeat_mset a A = repeat_mset b A ⟷ a = b ∨ A = {#}"
by (auto simp: multiset_eq_iff)

lemma repeat_mset_eq_empty_iff: "repeat_mset n A = {#} ⟷ n = 0 ∨ A = {#}"
by (cases n) auto

lemma image_replicate_mset [simp]:
"image_mset f (replicate_mset n a) = replicate_mset n (f a)"
by (induct n) simp_all

subsection ‹Big operators›

locale comm_monoid_mset = comm_monoid
begin

interpretation comp_fun_commute f
by standard (simp add: fun_eq_iff left_commute)

interpretation comp?: comp_fun_commute "f ∘ g"
by (fact comp_comp_fun_commute)

context
begin

definition F :: "'a multiset ⇒ 'a"
where eq_fold: "F M = fold_mset f ❙1 M"

lemma empty [simp]: "F {#} = ❙1"

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

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

lemma add_mset [simp]: "F (add_mset x N) = x ❙* F N"

lemma insert [simp]:
shows "F (image_mset g (add_mset x A)) = g x ❙* F (image_mset g A)"

lemma remove:
assumes "x ∈# A"
shows "F A = x ❙* F (A - {#x#})"
using multi_member_split[OF assms] by auto

lemma neutral:
"∀x∈#A. x = ❙1 ⟹ F A = ❙1"
by (induct A) simp_all

lemma neutral_const [simp]:
"F (image_mset (λ_. ❙1) A) = ❙1"

private lemma F_image_mset_product:
"F {#g x j ❙* F {#g i j. i ∈# A#}. j ∈# B#} =
F (image_mset (g x) B) ❙* F {#F {#g i j. i ∈# A#}. j ∈# B#}"
by (induction B) (simp_all add: left_commute semigroup.assoc semigroup_axioms)

lemma commute:
"F (image_mset (λi. F (image_mset (g i) B)) A) =
F (image_mset (λj. F (image_mset (λi. g i j) A)) B)"
apply (induction A, simp)
apply (induction B, auto simp add: F_image_mset_product ac_simps)
done

lemma distrib: "F (image_mset (λx. g x ❙* h x) A) = F (image_mset g A) ❙* F (image_mset h A)"
by (induction A) (auto simp: ac_simps)

lemma union_disjoint:
"A ∩# B = {#} ⟹ F (image_mset g (A ∪# B)) = F (image_mset g A) ❙* F (image_mset g B)"
by (induction A) (auto simp: ac_simps)

end
end

lemma comp_fun_commute_plus_mset[simp]: "comp_fun_commute (op + :: 'a multiset ⇒ _ ⇒ _)"

lemma in_mset_fold_plus_iff[iff]: "x ∈# fold_mset (op +) M NN ⟷ x ∈# M ∨ (∃N. N ∈# NN ∧ x ∈# N)"
by (induct NN) auto

begin

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

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

lemma sum_unfold_sum_mset:
"sum f A = sum_mset (image_mset f (mset_set A))"
by (cases "finite A") (induct A rule: finite_induct, simp_all)

lemma sum_mset_delta: "sum_mset (image_mset (λx. if x = y then c else 0) A) = c * count A y"
by (induction A) simp_all

lemma sum_mset_delta': "sum_mset (image_mset (λx. if y = x then c else 0) A) = c * count A y"
by (induction A) simp_all

end

lemma of_nat_sum_mset [simp]:
"of_nat (sum_mset M) = sum_mset (image_mset of_nat M)"
by (induction M) auto

lemma sum_mset_0_iff [simp]:
⟷ (∀x ∈ set_mset M. x = 0)"
by(induction M) auto

lemma sum_mset_diff:
fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
shows "N ⊆# M ⟹ sum_mset (M - N) = sum_mset M - sum_mset N"

lemma size_eq_sum_mset: "size M = sum_mset (image_mset (λ_. 1) M)"
proof (induct M)
case empty then show ?case by simp
next
case (add x M) then show ?case
by (cases "x ∈ set_mset M")
sum.remove)
qed

lemma size_mset_set [simp]: "size (mset_set A) = card A"
by (simp only: size_eq_sum_mset card_eq_sum sum_unfold_sum_mset)

lemma sum_mset_sum_list: "sum_mset (mset xs) = sum_list xs"
by (induction xs) auto

syntax (ASCII)
"_sum_mset_image" :: "pttrn ⇒ 'b set ⇒ 'a ⇒ 'a::comm_monoid_add"  ("(3SUM _:#_. _)" [0, 51, 10] 10)
syntax
"_sum_mset_image" :: "pttrn ⇒ 'b set ⇒ 'a ⇒ 'a::comm_monoid_add"  ("(3∑_∈#_. _)" [0, 51, 10] 10)
translations
"∑i ∈# A. b" ⇌ "CONST sum_mset (CONST image_mset (λi. b) A)"

lemma sum_mset_distrib_left:
fixes f :: "'a ⇒ 'b::semiring_0"
shows "c * (∑x ∈# M. f x) = (∑x ∈# M. c * f(x))"
by (induction M) (simp_all add: distrib_left)

lemma sum_mset_distrib_right:
fixes f :: "'a ⇒ 'b::semiring_0"
shows "(∑b ∈# B. f b) * a = (∑b ∈# B. f b * a)"
by (induction B) (auto simp: distrib_right)

lemma sum_mset_constant [simp]:
fixes y :: "'b::semiring_1"
shows ‹(∑x∈#A. y) = of_nat (size A) * y›
by (induction A) (auto simp: algebra_simps)

assumes "⋀i. i ∈# K ⟹ f i ≤ g i"
shows "sum_mset (image_mset f K) ≤ sum_mset (image_mset g K)"

lemma sum_mset_product:
fixes f :: "'a::{comm_monoid_add,times} ⇒ 'b::semiring_0"
shows "(∑i ∈# A. f i) * (∑i ∈# B. g i) = (∑i∈#A. ∑j∈#B. f i * g j)"
by (subst sum_mset.commute) (simp add: sum_mset_distrib_left sum_mset_distrib_right)

abbreviation Union_mset :: "'a multiset multiset ⇒ 'a multiset"  ("⋃#_" [900] 900)
where "⋃# MM ≡ sum_mset MM" ― ‹FIXME ambiguous notation --
could likewise refer to ‹⨆#››

lemma set_mset_Union_mset[simp]: "set_mset (⋃# MM) = (⋃M ∈ set_mset MM. set_mset M)"
by (induct MM) auto

lemma in_Union_mset_iff[iff]: "x ∈# ⋃# MM ⟷ (∃M. M ∈# MM ∧ x ∈# M)"
by (induct MM) auto

lemma count_sum:
"count (sum f A) x = sum (λa. count (f a) x) A"
by (induct A rule: infinite_finite_induct) simp_all

lemma sum_eq_empty_iff:
assumes "finite A"
shows "sum f A = {#} ⟷ (∀a∈A. f a = {#})"
using assms by induct simp_all

lemma Union_mset_empty_conv[simp]: "⋃# M = {#} ⟷ (∀i∈#M. i = {#})"
by (induction M) auto

context comm_monoid_mult
begin

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

lemma prod_mset_empty:
"prod_mset {#} = 1"
by (fact prod_mset.empty)

lemma prod_mset_singleton:
"prod_mset {#x#} = x"
by (fact prod_mset.singleton)

lemma prod_mset_Un:
"prod_mset (A + B) = prod_mset A * prod_mset B"
by (fact prod_mset.union)

lemma prod_mset_replicate_mset [simp]:
"prod_mset (replicate_mset n a) = a ^ n"
by (induct n) simp_all

lemma prod_unfold_prod_mset:
"prod f A = prod_mset (image_mset f (mset_set A))"
by (cases "finite A") (induct A rule: finite_induct, simp_all)

lemma prod_mset_multiplicity:
"prod_mset M = prod (λx. x ^ count M x) (set_mset M)"
by (simp add: fold_mset_def prod.eq_fold prod_mset.eq_fold funpow_times_power comp_def)

lemma prod_mset_delta: "prod_mset (image_mset (λx. if x = y then c else 1) A) = c ^ count A y"
by (induction A) simp_all

lemma prod_mset_delta': "prod_mset (image_mset (λx. if y = x then c else 1) A) = c ^ count A y"
by (induction A) simp_all

end

syntax (ASCII)
"_prod_mset_image" :: "pttrn ⇒ 'b set ⇒ 'a ⇒ 'a::comm_monoid_mult"  ("(3PROD _:#_. _)" [0, 51, 10] 10)
syntax
"_prod_mset_image" :: "pttrn ⇒ 'b set ⇒ 'a ⇒ 'a::comm_monoid_mult"  ("(3∏_∈#_. _)" [0, 51, 10] 10)
translations
"∏i ∈# A. b" ⇌ "CONST prod_mset (CONST image_mset (λi. b) A)"

lemma (in comm_monoid_mult) prod_mset_subset_imp_dvd:
assumes "A ⊆# B"
shows   "prod_mset A dvd prod_mset B"
proof -
from assms have "B = (B - A) + A" by (simp add: subset_mset.diff_add)
also have "prod_mset … = prod_mset (B - A) * prod_mset A" by simp
also have "prod_mset A dvd …" by simp
finally show ?thesis .
qed

lemma (in comm_monoid_mult) dvd_prod_mset:
assumes "x ∈# A"
shows "x dvd prod_mset A"
using assms prod_mset_subset_imp_dvd [of "{#x#}" A] by simp

lemma (in semidom) prod_mset_zero_iff [iff]:
"prod_mset A = 0 ⟷ 0 ∈# A"
by (induct A) auto

lemma (in semidom_divide) prod_mset_diff:
assumes "B ⊆# A" and "0 ∉# B"
shows "prod_mset (A - B) = prod_mset A div prod_mset B"
proof -
from assms obtain C where "A = B + C"
with assms show ?thesis by simp
qed

lemma (in semidom_divide) prod_mset_minus:
assumes "a ∈# A" and "a ≠ 0"
shows "prod_mset (A - {#a#}) = prod_mset A div a"
using assms prod_mset_diff [of "{#a#}" A] by auto

lemma (in algebraic_semidom) is_unit_prod_mset_iff:
"is_unit (prod_mset A) ⟷ (∀x ∈# A. is_unit x)"
by (induct A) (auto simp: is_unit_mult_iff)

lemma (in normalization_semidom) normalize_prod_mset:
"normalize (prod_mset A) = prod_mset (image_mset normalize A)"
by (induct A) (simp_all add: normalize_mult)

lemma (in normalization_semidom) normalized_prod_msetI:
assumes "⋀a. a ∈# A ⟹ normalize a = a"
shows "normalize (prod_mset A) = prod_mset A"
proof -
from assms have "image_mset normalize A = A"
by (induct A) simp_all
then show ?thesis by (simp add: normalize_prod_mset)
qed

lemma prod_mset_prod_list: "prod_mset (mset xs) = prod_list xs"
by (induct xs) auto

subsection ‹Alternative representations›

subsubsection ‹Lists›

context linorder
begin

lemma mset_insort [simp]:
"mset (insort_key k x xs) = add_mset x (mset xs)"
by (induct xs) simp_all

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

text ‹
This lemma shows which properties suffice to show that a function
‹f› with ‹f xs = ys› behaves like sort.
›

lemma properties_for_sort_key:
assumes "mset ys = mset xs"
and "⋀k. k ∈ set ys ⟹ filter (λx. f k = f x) ys = filter (λx. 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
"∀k ∈ set ys. filter (λx. f k = f x) (remove1 x ys) = filter (λx. f k = f x) xs"
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 ∈# mset ys"
by auto
then have "x ∈ set ys"
by simp
ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
qed

lemma properties_for_sort:
assumes multiset: "mset ys = mset xs"
and "sorted ys"
shows "sort xs = ys"
proof (rule properties_for_sort_key)
from multiset show "mset ys = mset xs" .
from ‹sorted ys› show "sorted (map (λx. x) ys)" by simp
from multiset have "length (filter (λy. k = y) ys) = length (filter (λx. k = x) xs)" for k
by (rule mset_eq_length_filter)
then have "replicate (length (filter (λy. k = y) ys)) k =
replicate (length (filter (λx. k = x) xs)) k" for k
by simp
then show "k ∈ set ys ⟹ filter (λy. k = y) ys = filter (λx. k = x) xs" for k
qed

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

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

lemma sort_key_by_quicksort:
"sort_key f xs = sort_key f [x←xs. f x < f (xs ! (length xs div 2))]
@ [x←xs. f x = f (xs ! (length xs div 2))]
@ sort_key f [x←xs. f x > f (xs ! (length xs div 2))]" (is "sort_key f ?lhs = ?rhs")
proof (rule properties_for_sort_key)
show "mset ?rhs = mset ?lhs"
by (rule multiset_eqI) (auto simp add: mset_filter)
show "sorted (map f ?rhs)"
by (auto simp add: sorted_append intro: sorted_map_same)
next
fix l
assume "l ∈ set ?rhs"
let ?pivot = "f (xs ! (length xs div 2))"
have *: "⋀x. f l = f x ⟷ f x = f l" by auto
have "[x ← sort_key f xs . f x = f l] = [x ← xs. f x = f l]"
unfolding filter_sort by (rule properties_for_sort_key) (auto intro: sorted_map_same)
with * have **: "[x ← sort_key f xs . f l = f x] = [x ← xs. f l = f x]" by simp
have "⋀x P. P (f x) ?pivot ∧ f l = f x ⟷ P (f l) ?pivot ∧ f l = f x" by auto
then have "⋀P. [x ← sort_key f xs . P (f x) ?pivot ∧ f l = f x] =
[x ← sort_key f xs. P (f l) ?pivot ∧ f l = f x]" by simp
note *** = this [of "op <"] this [of "op >"] this [of "op ="]
show "[x ← ?rhs. f l = f x] = [x ← ?lhs. f l = f x]"
proof (cases "f l" ?pivot rule: linorder_cases)
case less
then have "f l ≠ ?pivot" and "¬ f l > ?pivot" by auto
with less show ?thesis
by (simp add: filter_sort [symmetric] ** ***)
next
case equal then show ?thesis
next
case greater
then have "f l ≠ ?pivot" and "¬ f l < ?pivot" by auto
with greater show ?thesis
by (simp add: filter_sort [symmetric] ** ***)
qed
qed

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

text ‹A stable parametrized quicksort›

definition part :: "('b ⇒ 'a) ⇒ 'a ⇒ 'b list ⇒ 'b list × 'b list × 'b list" where
"part f pivot xs = ([x ← xs. f x < pivot], [x ← xs. f x = pivot], [x ← 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
[] ⇒ []
| [x] ⇒ xs
| [x, y] ⇒ (if f x ≤ f y then xs else [y, x])
| _ ⇒
let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
in sort_key f lts @ eqs @ sort_key f gts)"
proof (cases xs)
case Nil then show ?thesis by simp
next
case (Cons _ ys) note hyps = Cons show ?thesis
proof (cases ys)
case Nil with hyps show ?thesis by simp
next
case (Cons _ zs) note hyps = hyps Cons show ?thesis
proof (cases zs)
case Nil with hyps show ?thesis by auto
next
case Cons
from sort_key_by_quicksort [of f xs]
have "sort_key f xs = (let (lts, eqs, gts) = part f (f (xs ! (length xs div 2))) xs
in sort_key f lts @ eqs @ sort_key f gts)"
by (simp only: split_def Let_def part_def fst_conv snd_conv)
with hyps Cons show ?thesis by (simp only: list.cases)
qed
qed
qed

end

hide_const (open) part

lemma mset_remdups_subset_eq: "mset (remdups xs) ⊆# mset xs"
by (induct xs) (auto intro: subset_mset.order_trans)

lemma mset_update:
"i < length ls ⟹ mset (ls[i := v]) = add_mset v (mset ls - {#ls ! i#})"
proof (induct ls arbitrary: i)
case Nil then show ?case by simp
next
case (Cons x xs)
show ?case
proof (cases i)
case 0 then show ?thesis by simp
next
case (Suc i')
with Cons show ?thesis
by (cases ‹x = xs ! i'›) auto
qed
qed

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

subsection ‹The multiset order›

subsubsection ‹Well-foundedness›

definition mult1 :: "('a × 'a) set ⇒ ('a multiset × 'a multiset) set" where
"mult1 r = {(N, M). ∃a M0 K. M = add_mset a M0 ∧ N = M0 + K ∧
(∀b. b ∈# K ⟶ (b, a) ∈ r)}"

definition mult :: "('a × 'a) set ⇒ ('a multiset × 'a multiset) set" where
"mult r = (mult1 r)⇧+"

lemma mult1I:
assumes "M = add_mset a M0" and "N = M0 + K" and "⋀b. b ∈# K ⟹ (b, a) ∈ r"
shows "(N, M) ∈ mult1 r"
using assms unfolding mult1_def by blast

lemma mult1E:
assumes "(N, M) ∈ mult1 r"
obtains a M0 K where "M = add_mset a M0" "N = M0 + K" "⋀b. b ∈# K ⟹ (b, a) ∈ r"
using assms unfolding mult1_def by blast

lemma mono_mult1:
assumes "r ⊆ r'" shows "mult1 r ⊆ mult1 r'"
unfolding mult1_def using assms by blast

lemma mono_mult:
assumes "r ⊆ r'" shows "mult r ⊆ mult r'"
unfolding mult_def using mono_mult1[OF assms] trancl_mono by blast

lemma not_less_empty [iff]: "(M, {#}) ∉ mult1 r"

assumes mult1: "(N, add_mset a M0) ∈ mult1 r"
shows
"(∃M. (M, M0) ∈ mult1 r ∧ N = add_mset a M) ∨
(∃K. (∀b. b ∈# K ⟶ (b, a) ∈ r) ∧ N = M0 + K)"
proof -
let ?r = "λK a. ∀b. b ∈# K ⟶ (b, a) ∈ r"
let ?R = "λN M. ∃a M0 K. M = add_mset a M0 ∧ N = M0 + K ∧ ?r K a"
obtain a' M0' K where M0: "add_mset a M0 = add_mset a' M0'"
and N: "N = M0' + K"
and r: "?r K a'"
using mult1 unfolding mult1_def by auto
show ?thesis (is "?case1 ∨ ?case2")
proof -
from M0 consider "M0 = M0'" "a = a'"
| K' where "M0 = add_mset a' K'" "M0' = add_mset a K'"
then show ?thesis
proof cases
case 1
with N r have "?r K a ∧ N = M0 + K" by simp
then have ?case2 ..
then show ?thesis ..
next
case 2
from N 2(2) have n: "N = add_mset a (K' + K)" by simp
with r 2(1) have "?R (K' + K) M0" by blast
with n have ?case1 by (simp add: mult1_def)
then show ?thesis ..
qed
qed
qed

lemma all_accessible:
assumes "wf r"
shows "∀M. M ∈ Wellfounded.acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "Wellfounded.acc ?R"
{
fix M M0 a
assume M0: "M0 ∈ ?W"
and wf_hyp: "⋀b. (b, a) ∈ r ⟹ (∀M ∈ ?W. add_mset b M ∈ ?W)"
and acc_hyp: "∀M. (M, M0) ∈ ?R ⟶ add_mset a M ∈ ?W"
have "add_mset a M0 ∈ ?W"
proof (rule accI [of "add_mset a M0"])
fix N
assume "(N, add_mset a M0) ∈ ?R"
then consider M where "(M, M0) ∈ ?R" "N = add_mset a M"
| K where "∀b. b ∈# K ⟶ (b, a) ∈ r" "N = M0 + K"
then show "N ∈ ?W"
proof cases
case 1
from acc_hyp have "(M, M0) ∈ ?R ⟶ add_mset a M ∈ ?W" ..
from this and ‹(M, M0) ∈ ?R› have "add_mset a M ∈ ?W" ..
then show "N ∈ ?W" by (simp only: ‹N = add_mset a M›)
next
case 2
from this(1) have "M0 + K ∈ ?W"
proof (induct K)
case empty
from M0 show "M0 + {#} ∈ ?W" by simp
next
from add.prems have "(x, a) ∈ r" by simp
with wf_hyp have "∀M ∈ ?W. add_mset x M ∈ ?W" by blast
moreover from add have "M0 + K ∈ ?W" by simp
ultimately have "add_mset x (M0 + K) ∈ ?W" ..
then show "M0 + (add_mset x K) ∈ ?W" by simp
qed
then show "N ∈ ?W" by (simp only: 2(2))
qed
qed
} note tedious_reasoning = this

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

fix M a assume "M ∈ ?W"
from ‹wf r› have "∀M ∈ ?W. add_mset a M ∈ ?W"
proof induct
fix a
assume r: "⋀b. (b, a) ∈ r ⟹ (∀M ∈ ?W. add_mset b M ∈ ?W)"
show "∀M ∈ ?W. add_mset a M ∈ ?W"
proof
fix M assume "M ∈ ?W"
then show "add_mset a M ∈ ?W"
by (rule acc_induct) (rule tedious_reasoning [OF _ r])
qed
qed
from this and ‹M ∈ ?W› show "add_mset a M ∈ ?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:
assumes
trans: "trans r" and
MN: "(M, N) ∈ mult r"
shows "∃I J K. N = I + J ∧ M = I + K ∧ J ≠ {#} ∧ (∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ r)"
using MN unfolding mult_def mult1_def
proof (induction rule: converse_trancl_induct)
case (base y)
then show ?case by force
next
case (step y z) note yz = this(1) and zN = this(2) and N_decomp = this(3)
obtain I J K where
N: "N = I + J" "z = I + K" "J ≠ {#}" "∀k∈#K. ∃j∈#J. (k, j) ∈ r"
using N_decomp by blast
obtain a M0 K' where
z: "z = add_mset a M0" and y: "y = M0 + K'" and K: "∀b. b ∈# K' ⟶ (b, a) ∈ r"
using yz by blast
show ?case
proof (cases "a ∈# K")
case True
moreover have "∃j∈#J. (k, j) ∈ r" if "k ∈# K'" for k
using K N trans True by (meson that transE)
ultimately show ?thesis
by (rule_tac x = I in exI, rule_tac x = J in exI, rule_tac x = "(K - {#a#}) + K'" in exI)
(use z y N in ‹auto simp del: subset_mset.add_diff_assoc2 dest: in_diffD›)
next
case False
then have "a ∈# I" by (metis N(2) union_iff union_single_eq_member z)
moreover have "M0 = I + K - {#a#}"
using N(2) z by force
ultimately show ?thesis
by (rule_tac x = "I - {#a#}" in exI, rule_tac x = "add_mset a J" in exI,
rule_tac x = "K + K'" in exI)
(use z y N False K in ‹auto simp: add.assoc›)
qed
qed

lemma one_step_implies_mult:
assumes
"J ≠ {#}" and
"∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ r"
shows "(I + K, I + J) ∈ mult r"
using assms
proof (induction "size J" arbitrary: I J K)
case 0
then show ?case by auto
next
case (Suc n) note IH = this(1) and size_J = this(2)[THEN sym]
obtain J' a where J: "J = add_mset a J'"
using size_J by (blast dest: size_eq_Suc_imp_eq_union)
show ?case
proof (cases "J' = {#}")
case True
then show ?thesis
using J Suc by (fastforce simp add: mult_def mult1_def)
next
case [simp]: False
have K: "K = {#x ∈# K. (x, a) ∈ r#} + {#x ∈# K. (x, a) ∉ r#}"
by (rule multiset_partition)
have "(I + K, (I + {# x ∈# K. (x, a) ∈ r #}) + J') ∈ mult r"
using IH[of J' "{# x ∈# K. (x, a) ∉ r#}" "I + {# x ∈# K. (x, a) ∈ r#}"]
J Suc.prems K size_J by (auto simp: ac_simps)
moreover have "(I + {#x ∈# K. (x, a) ∈ r#} + J', I + J) ∈ mult r"
by (fastforce simp: J mult1_def mult_def)
ultimately show ?thesis
unfolding mult_def by simp
qed
qed

subsection ‹The multiset extension is cancellative for multiset union›

lemma mult_cancel:
assumes "trans s" and "irrefl s"
shows "(X + Z, Y + Z) ∈ mult s ⟷ (X, Y) ∈ mult s" (is "?L ⟷ ?R")
proof
assume ?L thus ?R
proof (induct Z)
obtain X' Y' Z' where *: "add_mset z X + Z = Z' + X'" "add_mset z Y + Z = Z' + Y'" "Y' ≠ {#}"
"∀x ∈ set_mset X'. ∃y ∈ set_mset Y'. (x, y) ∈ s"
using mult_implies_one_step[OF `trans s` add(2)] by auto
consider Z2 where "Z' = add_mset z Z2" | X2 Y2 where "X' = add_mset z X2" "Y' = add_mset z Y2"
thus ?case
proof (cases)
case 1 thus ?thesis using * one_step_implies_mult[of Y' X' s Z2]
next
case 2 then obtain y where "y ∈ set_mset Y2" "(z, y) ∈ s" using *(4) `irrefl s`
by (auto simp: irrefl_def)
moreover from this transD[OF `trans s` _ this(2)]
have "x' ∈ set_mset X2 ⟹ ∃y ∈ set_mset Y2. (x', y) ∈ s" for x'
using 2 *(4)[rule_format, of x'] by auto
ultimately show ?thesis using  * one_step_implies_mult[of Y2 X2 s Z'] 2
qed
qed auto
next
assume ?R then obtain I J K
where "Y = I + J" "X = I + K" "J ≠ {#}" "∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ s"
using mult_implies_one_step[OF `trans s`] by blast
thus ?L using one_step_implies_mult[of J K s "I + Z"] by (auto simp: ac_simps)
qed

lemma mult_cancel_max:
assumes "trans s" and "irrefl s"
shows "(X, Y) ∈ mult s ⟷ (X - X ∩# Y, Y - X ∩# Y) ∈ mult s" (is "?L ⟷ ?R")
proof -
have "X - X ∩# Y + X ∩# Y = X" "Y - X ∩# Y + X ∩# Y = Y" by (auto simp: count_inject[symmetric])
thus ?thesis using mult_cancel[OF assms, of "X - X ∩# Y"  "X ∩# Y" "Y - X ∩# Y"] by auto
qed

subsection ‹Quasi-executable version of the multiset extension›

text ‹
Predicate variants of ‹mult› and the reflexive closure of ‹mult›, which are
executable whenever the given predicate ‹P› is. Together with the standard
code equations for ‹op ∩#› and ‹op -› this should yield quadratic
(with respect to calls to ‹P›) implementations of ‹multp› and ‹multeqp›.
›

definition multp :: "('a ⇒ 'a ⇒ bool) ⇒ 'a multiset ⇒ 'a multiset ⇒ bool" where
"multp P N M =
(let Z = M ∩# N; X = M - Z in
X ≠ {#} ∧ (let Y = N - Z in (∀y ∈ set_mset Y. ∃x ∈ set_mset X. P y x)))"

definition multeqp :: "('a ⇒ 'a ⇒ bool) ⇒ 'a multiset ⇒ 'a multiset ⇒ bool" where
"multeqp P N M =
(let Z = M ∩# N; X = M - Z; Y = N - Z in
(∀y ∈ set_mset Y. ∃x ∈ set_mset X. P y x))"

lemma multp_iff:
assumes "irrefl R" and "trans R" and [simp]: "⋀x y. P x y ⟷ (x, y) ∈ R"
shows "multp P N M ⟷ (N, M) ∈ mult R" (is "?L ⟷ ?R")
proof -
have *: "M ∩# N + (N - M ∩# N) = N" "M ∩# N + (M - M ∩# N) = M"
"(M - M ∩# N) ∩# (N - M ∩# N) = {#}" by (auto simp: count_inject[symmetric])
show ?thesis
proof
assume ?L thus ?R
using one_step_implies_mult[of "M - M ∩# N" "N - M ∩# N" R "M ∩# N"] *
by (auto simp: multp_def Let_def)
next
{ fix I J K :: "'a multiset" assume "(I + J) ∩# (I + K) = {#}"
then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty)
} note [dest!] = this
assume ?R thus ?L
using mult_implies_one_step[OF assms(2), of "N - M ∩# N" "M - M ∩# N"]
mult_cancel_max[OF assms(2,1), of "N" "M"] * by (auto simp: multp_def)
qed
qed

lemma multeqp_iff:
assumes "irrefl R" and "trans R" and "⋀x y. P x y ⟷ (x, y) ∈ R"
shows "multeqp P N M ⟷ (N, M) ∈ (mult R)⇧="
proof -
{ assume "N ≠ M" "M - M ∩# N = {#}"
then obtain y where "count N y ≠ count M y" by (auto simp: count_inject[symmetric])
then have "∃y. count M y < count N y" using `M - M ∩# N = {#}`
by (auto simp: count_inject[symmetric] dest!: le_neq_implies_less fun_cong[of _ _ y])
}
then have "multeqp P N M ⟷ multp P N M ∨ N = M"
by (auto simp: multeqp_def multp_def Let_def in_diff_count)
thus ?thesis using multp_iff[OF assms] by simp
qed

subsubsection ‹Partial-order properties›

lemma (in preorder) mult1_lessE:
assumes "(N, M) ∈ mult1 {(a, b). a < b}"
obtains a M0 K where "M = add_mset a M0" "N = M0 + K"
"a ∉# K" "⋀b. b ∈# K ⟹ b < a"
proof -
from assms obtain a M0 K where "M = add_mset a M0" "N = M0 + K" and
*: "b ∈# K ⟹ b < a" for b by (blast elim: mult1E)
moreover from * [of a] have "a ∉# K" by auto
ultimately show thesis by (auto intro: that)
qed

instantiation multiset :: (preorder) order
begin

definition less_multiset :: "'a multiset ⇒ 'a multiset ⇒ bool"
where "M' < M ⟷ (M', M) ∈ mult {(x', x). x' < x}"

definition less_eq_multiset :: "'a multiset ⇒ 'a multiset ⇒ bool"
where "less_eq_multiset M' M ⟷ M' < M ∨ M' = M"

instance
proof -
have irrefl: "¬ M < M" for M :: "'a multiset"
proof
assume "M < M"
then have MM: "(M, M) ∈ mult {(x, y). x < y}" by (simp add: less_multiset_def)
have "trans {(x'::'a, x). x' < x}"
by (metis (mono_tags, lifting) case_prodD case_prodI less_trans mem_Collect_eq transI)
moreover note MM
ultimately have "∃I J K. M = I + J ∧ M = I + K
∧ J ≠ {#} ∧ (∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(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 ≠ {#}" and "(∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(x, y). x < y})" by blast
then have *: "K ≠ {#}" and **: "∀k∈set_mset K. ∃j∈set_mset K. k < j" by auto
have "finite (set_mset K)" by simp
moreover note **
ultimately have "set_mset K = {}"
by (induct rule: finite_induct) (auto intro: order_less_trans)
with * show False by simp
qed
have trans: "K < M ⟹ M < N ⟹ K < N" for K M N :: "'a multiset"
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
show "OFCLASS('a multiset, order_class)"
by standard (auto simp add: less_eq_multiset_def irrefl dest: trans)
qed
end ― ‹FIXME avoid junk stemming from type class interpretation›

lemma mset_le_irrefl [elim!]:
fixes M :: "'a::preorder multiset"
shows "M < M ⟹ R"
by simp

subsubsection ‹Monotonicity of multiset union›

lemma mult1_union: "(B, D) ∈ mult1 r ⟹ (C + B, C + D) ∈ mult1 r"
by (force simp: mult1_def)

lemma union_le_mono2: "B < D ⟹ C + B < C + (D::'a::preorder 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_le_mono1: "B < D ⟹ B + C < D + (C::'a::preorder multiset)"
apply (subst add.commute [of B C])
apply (subst add.commute [of D C])
apply (erule union_le_mono2)
done

lemma union_less_mono:
fixes A B C D :: "'a::preorder multiset"
shows "A < C ⟹ B < D ⟹ A + B < C + D"
by (blast intro!: union_le_mono1 union_le_mono2 less_trans)

begin
instance
by standard (auto simp add: less_eq_multiset_def intro: union_le_mono2)
end

subsubsection ‹Termination proofs with multiset orders›

lemma multi_member_skip: "x ∈# XS ⟹ x ∈# {# y #} + XS"
and multi_member_this: "x ∈# {# x #} + XS"
and multi_member_last: "x ∈# {# x #}"
by auto

definition "ms_strict = mult pair_less"
definition "ms_weak = ms_strict ∪ Id"

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

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

lemma wmsI:
"(set_mset A, set_mset B) ∈ max_strict ∨ A = {#} ∧ B = {#}
⟹ (Z + A, Z + B) ∈ 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:  "⟦(x,y) ∈ pair_leq; pw_leq X Y ⟧ ⟹ pw_leq ({#x#} + X) ({#y#} + Y)"

lemma pw_leq_lstep:
"(x, y) ∈ pair_leq ⟹ pw_leq {#x#} {#y#}"
by (drule pw_leq_step) (rule pw_leq_empty, simp)

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

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

lemma empty_neutral: "{#} + x = x" "x + {#} = x"
and nonempty_plus: "{# x #} + rs ≠ {#}"
and nonempty_single: "{# x #} ≠ {#}"
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 add_mset}, T --> msetT T --> msetT T) \$ x \$
Const (@{const_abbrev Mempty}, msetT T)
| mk_mset T (x :: xs) =
Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
mk_mset T [x] \$ mk_mset T xs

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

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

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

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

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

subsection ‹Legacy theorem bindings›

lemmas multi_count_eq = multiset_eq_iff [symmetric]

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

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

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

lemmas union_ac = union_assoc union_commute union_lcomm add_mset_commute

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

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

lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y ⟹ X = Y"

lemma mset_subset_trans: "(M::'a multiset) ⊂# K ⟹ K ⊂# N ⟹ M ⊂# N"
by (fact subset_mset.less_trans)

lemma multiset_inter_commute: "A ∩# B = B ∩# A"
by (fact subset_mset.inf.commute)

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

lemma multiset_inter_left_commute: "A ∩# (B ∩# C) = B ∩# (A ∩# C)"
by (fact subset_mset.inf.left_commute)

lemmas multiset_inter_ac =
multiset_inter_commute
multiset_inter_assoc
multiset_inter_left_commute

lemma mset_le_not_refl: "¬ M < (M::'a::preorder multiset)"
by (fact less_irrefl)

lemma mset_le_trans: "K < M ⟹ M < N ⟹ K < (N::'a::preorder multiset)"
by (fact less_trans)

lemma mset_le_not_sym: "M < N ⟹ ¬ N < (M::'a::preorder multiset)"
by (fact less_not_sym)

lemma mset_le_asym: "M < N ⟹ (¬ P ⟹ N < (M::'a::preorder multiset)) ⟹ P"
by (fact less_asym)

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

subsection ‹Naive implementation using lists›

code_datatype mset

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

lemma [code]: "add_mset x (mset xs) = mset (x # xs)"
by simp

lemma [code]: "Multiset.is_empty (mset xs) ⟷ List.null xs"

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

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

lemma [code]: "filter_mset f (mset xs) = mset (filter f xs)"

lemma [code]: "mset xs - mset ys = mset (fold remove1 ys xs)"

lemma [code]:
"mset xs ∩# mset ys =
mset (snd (fold (λx (ys, zs).
if x ∈ set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
proof -
have "⋀zs. mset (snd (fold (λx (ys, zs).
if x ∈ set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
(mset xs ∩# mset ys) + mset zs"
by (induct xs arbitrary: ys)
then show ?thesis by simp
qed

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

declare in_multiset_in_set [code_unfold]

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

declare set_mset_mset [code]

declare sorted_list_of_multiset_mset [code]

lemma [code]: ― ‹not very efficient, but representation-ignorant!›
"mset_set A = mset (sorted_list_of_set A)"
apply (cases "finite A")
apply simp_all
apply (induct A rule: finite_induct)
apply simp_all
done

declare size_mset [code]

fun subset_eq_mset_impl :: "'a list ⇒ 'a list ⇒ bool option" where
"subset_eq_mset_impl [] ys = Some (ys ≠ [])"
| "subset_eq_mset_impl (Cons x xs) ys = (case List.extract (op = x) ys of
None ⇒ None
| Some (ys1,_,ys2) ⇒ subset_eq_mset_impl xs (ys1 @ ys2))"

lemma subset_eq_mset_impl: "(subset_eq_mset_impl xs ys = None ⟷ ¬ mset xs ⊆# mset ys) ∧
(subset_eq_mset_impl xs ys = Some True ⟷ mset xs ⊂# mset ys) ∧
(subset_eq_mset_impl xs ys = Some False ⟶ mset xs = mset ys)"
proof (induct xs arbitrary: ys)
case (Nil ys)
show ?case by (auto simp: subset_mset.zero_less_iff_neq_zero)
next
case (Cons x xs ys)
show ?case
proof (cases "List.extract (op = x) ys")
case None
hence x: "x ∉ set ys" by (simp add: extract_None_iff)
{
assume "mset (x # xs) ⊆# mset ys"
from set_mset_mono[OF this] x have False by simp
} note nle = this
moreover
{
assume "mset (x # xs) ⊂# mset ys"
hence "mset (x # xs) ⊆# mset ys" by auto
from nle[OF this] have False .
}
ultimately show ?thesis using None by auto
next
case (Some res)
obtain ys1 y ys2 where res: "res = (ys1,y,ys2)" by (cases res, auto)
note Some = Some[unfolded res]
from extract_SomeE[OF Some] have "ys = ys1 @ x # ys2" by simp
hence id: "mset ys = add_mset x (mset (ys1 @ ys2))"
by auto
show ?thesis unfolding subset_eq_mset_impl.simps
unfolding Some option.simps split
unfolding id
using Cons[of "ys1 @ ys2"]
unfolding subset_mset_def subseteq_mset_def by auto
qed
qed

lemma [code]: "mset xs ⊆# mset ys ⟷ subset_eq_mset_impl xs ys ≠ None"
using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)

lemma [code]: "mset xs ⊂# mset ys ⟷ subset_eq_mset_impl xs ys = Some True"
using subset_eq_mset_impl[of xs ys] by (cases "subset_eq_mset_impl xs ys", auto)

instantiation multiset :: (equal) equal
begin

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

instance

end

lemma [code]: "sum_mset (mset xs) = sum_list xs"
by (induct xs) simp_all

lemma [code]: "prod_mset (mset xs) = fold times xs 1"
proof -
have "⋀x. fold times xs x = prod_mset (mset xs) * x"
by (induct xs) (simp_all add: ac_simps)
then show ?thesis by simp
qed

text ‹
and @{term "op <"} (multiset order).
›

text ‹Quickcheck generators›

definition (in term_syntax)
msetify :: "'a::typerep list × (unit ⇒ Code_Evaluation.term)
⇒ 'a multiset × (unit ⇒ Code_Evaluation.term)" where
[code_unfold]: "msetify xs = Code_Evaluation.valtermify mset {⋅} xs"

notation fcomp (infixl "∘>" 60)
notation scomp (infixl "∘→" 60)

instantiation multiset :: (random) random
begin

definition
"Quickcheck_Random.random i = Quickcheck_Random.random i ∘→ (λxs. Pair (msetify xs))"

instance ..

end

no_notation fcomp (infixl "∘>" 60)
no_notation scomp (infixl "∘→" 60)

instantiation multiset :: (full_exhaustive) full_exhaustive
begin

definition full_exhaustive_multiset :: "('a multiset × (unit ⇒ term) ⇒ (bool × term list) option) ⇒ natural ⇒ (bool × term list) option"
where
"full_exhaustive_multiset f i = Quickcheck_Exhaustive.full_exhaustive (λxs. f (msetify xs)) i"

instance ..

end

hide_const (open) msetify

subsection ‹BNF setup›

definition rel_mset where
"rel_mset R X Y ⟷ (∃xs ys. mset xs = X ∧ mset ys = Y ∧ list_all2 R xs ys)"

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

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

define xsa where "xsa = take j xs' @ drop (Suc j) xs'"
have "mset xs' = {#x#} + mset xsa"
unfolding xsa_def using j_len nth_j
append_take_drop_id mset.simps(2) mset_append)
hence ms_x: "mset xsa = mset xs"
then obtain ysa where
len_a: "length ysa = length xsa" and ms_a: "mset (zip xsa ysa) = mset (zip xs ys)"
using Cons.hyps(2) by blast

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

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

lemma ex_mset: "∃xs. mset xs = X"
by (induct X) (simp, metis mset.simps(2))

inductive pred_mset :: "('a ⇒ bool) ⇒ 'a multiset ⇒ bool"
where
"pred_mset P {#}"
| "⟦P a; pred_mset P M⟧ ⟹ pred_mset P (add_mset a M)"

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

inductive rel_mset'
where
Zero[intro]: "rel_mset' R {#} {#}"
| Plus[intro]: "⟦R a b; rel_mset' R M N⟧ ⟹ rel_mset' R (add_mset a M) (add_mset b N)"

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

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

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

lemma rel_mset'_imp_rel_mset: "rel_mset' R M N ⟹ rel_mset R M N"
by (induct rule: rel_mset'.induct) (auto simp: rel_mset_Zero rel_mset_Plus)

lemma rel_mset_size: "rel_mset R M N ⟹ size M = size N"
unfolding multiset.rel_compp_Grp Grp_def by auto

assumes empty: "P {#} {#}"
and addL: "⋀a M N. P M N ⟹ P (add_mset a M) N"
and addR: "⋀a M N. P M N ⟹ P M (add_mset a N)"
shows "P M N"
apply(induct N rule: multiset_induct)
apply(induct M rule: multiset_induct, rule empty, erule addL)
done

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

lemma msed_map_invL:
assumes "image_mset f (add_mset a M) = N"
shows "∃N1. N = add_mset (f a) N1 ∧ image_mset f M = N1"
proof -
have "f a ∈# N"
using assms multiset.set_map[of f "add_mset a M"] by auto
then obtain N1 where N: "N = add_mset (f a) N1" using multi_member_split by metis
have "image_mset f M = N1" using assms unfolding N by simp
thus ?thesis using N by blast
qed

lemma msed_map_invR:
assumes "image_mset f M = add_mset b N"
shows "∃M1 a. M = add_mset a M1 ∧ f a = b ∧ image_mset f M1 = N"
proof -
obtain a where a: "a ∈# M" and fa: "f a = b"
using multiset.set_map[of f M] unfolding assms
by (metis image_iff union_single_eq_member)
then obtain M1 where M: "M = add_mset a M1" using multi_member_split by metis
have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
thus ?thesis using M fa by blast
qed

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

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

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

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

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

subsection ‹Size setup›

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

setup ‹
BNF_LFP_Size.register_size_global @{type_name multiset} @{const_name size_multiset}