# 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

abbreviation Melem :: "'a ⇒ 'a multiset ⇒ bool"  (infix "∈#" 50)
where "a ∈# M ≡ 0 < count M a"

notation (ASCII)
Melem  ("(_/ :# _)" [50, 51] 50)  (* FIXME !? *)

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

lift_definition single :: "'a ⇒ 'a multiset" is "λa b. if b = a then 1 else 0"
by (rule only1_in_multiset)

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

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

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

subsection ‹Basic operations›

subsubsection ‹Union›

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

subsubsection ‹Difference›

instantiation multiset :: (type) comm_monoid_diff
begin

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

end

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

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

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

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

lemma diff_union_cancelL [simp]: "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: "x ∈# M ⟹ {#x#} + (M - {#x#}) = M"
by (clarsimp simp: multiset_eq_iff)

lemma insert_DiffM2 [simp]: "x ∈# M ⟹ M - {#x#} + {#x#} = M"
by (clarsimp simp: multiset_eq_iff)

lemma diff_union_swap: "a ≠ b ⟹ M - {#a#} + {#b#} = M + {#b#} - {#a#}"

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

subsubsection ‹Equality of multisets›

lemma single_not_empty [simp]: "{#a#} ≠ {#} ∧ {#} ≠ {#a#}"

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 = {#}"

lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} ⟷ False"

lemma diff_single_trivial: "¬ x ∈# M ⟹ M - {#x#} = M"

lemma diff_single_eq_union: "x ∈# M ⟹ M - {#x#} = N ⟷ M = N + {#x#}"
by auto

lemma union_single_eq_diff: "M + {#x#} = N ⟹ M = N - {#x#}"
by (auto dest: sym)

lemma union_single_eq_member: "M + {#x#} = N ⟹ x ∈# N"
by auto

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)

"M + {#a#} = N + {#b#} ⟷ M = N ∧ a = b ∨ M = N - {#a#} + {#b#} ∧ N = M - {#b#} + {#a#}"
(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 ∈# N + {#b#}" by (rule union_single_eq_member)
with False have "a ∈# N" by auto
moreover from ‹?lhs› have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)
moreover note False
ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)
qed
qed

lemma insert_noteq_member:
assumes BC: "B + {#b#} = C + {#c#}"
and bnotc: "b ≠ c"
shows "c ∈# B"
proof -
have "c ∈# C + {#c#}" by simp
have nc: "¬ c ∈# {#b#}" using bnotc by simp
then have "c ∈# B + {#b#}" using BC by simp
then show "c ∈# B" using nc by simp
qed

"(M + {#a#} = N + {#b#}) =
(M = N ∧ a = b ∨ (∃K. M = K + {#b#} ∧ N = K + {#a#}))"

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

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

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" where
"supseteq_mset A B == B ⊆# A"

abbreviation (input) supset_mset :: "'a multiset ⇒ 'a multiset ⇒ bool" where
"supset_mset A B == B ⊂# A"

notation (input)
subseteq_mset  (infix "≤#" 50) and
supseteq_mset  (infix "≥#" 50) and
supseteq_mset  (infix "⊇#" 50) and
supset_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)

lemma mset_less_eqI: "(⋀x. count A x ≤ count B x) ⟹ A ≤# B"

lemma mset_le_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 +" "op -" 0 "op ≤#" "op <#"
by standard (simp, fact mset_le_exists_conv)

lemma mset_le_mono_add_right_cancel [simp]: "(A::'a multiset) + C ≤# B + C ⟷ A ≤# B"

lemma mset_le_mono_add_left_cancel [simp]: "C + (A::'a multiset) ≤# C + B ⟷ A ≤# B"

lemma mset_le_mono_add: "(A::'a multiset) ≤# B ⟹ C ≤# D ⟹ A + C ≤# B + D"

lemma mset_le_add_left [simp]: "(A::'a multiset) ≤# A + B"
unfolding subseteq_mset_def by auto

lemma mset_le_add_right [simp]: "B ≤# (A::'a multiset) + B"
unfolding subseteq_mset_def by auto

lemma mset_le_single: "a ∈# B ⟹ {#a#} ≤# B"

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

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

lemma diff_le_self[simp]: "(M::'a multiset) - N ≤# M"

lemma mset_lessD: "A <# B ⟹ x ∈# A ⟹ x ∈# B"
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
apply (erule allE [where x = x])
apply auto
done

lemma mset_leD: "A ≤# B ⟹ x ∈# A ⟹ x ∈# B"
apply (clarsimp simp: subset_mset_def subseteq_mset_def)
apply (erule allE [where x = x])
apply auto
done

lemma mset_less_insertD: "(A + {#x#} <# 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: split_if_asm)
done

lemma mset_le_insertD: "(A + {#x#} ≤# B) ⟹ (x ∈# B ∧ A ≤# B)"
apply (rule conjI)
apply (force simp: subset_mset_def subseteq_mset_def split: split_if_asm)
done

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

lemma empty_le[simp]: "{#} ≤# A"
unfolding mset_le_exists_conv by auto

lemma le_empty[simp]: "(M ≤# {#}) = (M = {#})"
unfolding mset_le_exists_conv by auto

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

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

lemma mset_less_add_bothsides: "N + {#x#} <# M + {#x#} ⟹ N <# M"

lemma mset_less_empty_nonempty: "{#} <# S ⟷ S ≠ {#}"
by (auto simp: subset_mset_def subseteq_mset_def)

lemma mset_less_diff_self: "c ∈# B ⟹ B - {#c#} <# B"
by (auto simp: subset_mset_def subseteq_mset_def multiset_eq_iff)

subsubsection ‹Intersection›

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

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

lemma multiset_inter_single: "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"
by auto
then show "count (A + B - C) x = count (A - C + B) x"
by auto
qed

lemma empty_inter [simp]: "{#} #∩ M = {#}"

lemma inter_empty [simp]: "M #∩ {#} = {#}"

lemma inter_add_left1: "¬ x ∈# N ⟹ (M + {#x#}) #∩ N = M #∩ N"

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

lemma inter_add_right1: "¬ x ∈# N ⟹ N #∩ (M + {#x#}) = N #∩ M"

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

subsubsection ‹Bounded union›

definition sup_subset_mset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset"(infixl "#∪" 70)
where "sup_subset_mset A B = A + (B - A)"

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

lemma sup_subset_mset_count [simp]: "count (A #∪ B) x = max (count A x) (count B x)"

lemma empty_sup [simp]: "{#} #∪ M = M"

lemma sup_empty [simp]: "M #∪ {#} = M"

lemma sup_add_left1: "¬ x ∈# N ⟹ (M + {#x#}) #∪ N = (M #∪ N) + {#x#}"

lemma sup_add_left2: "x ∈# N ⟹ (M + {#x#}) #∪ N = (M #∪ (N - {#x#})) + {#x#}"

lemma sup_add_right1: "¬ x ∈# N ⟹ N #∪ (M + {#x#}) = (N #∪ M) + {#x#}"

lemma sup_add_right2: "x ∈# N ⟹ N #∪ (M + {#x#}) = ((N - {#x#}) #∪ M) + {#x#}"

subsubsection ‹Subset is an order›
interpretation subset_mset: order "op ≤#" "op <#" by unfold_locales auto

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)

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

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

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

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

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

lemma filter_inter_mset [simp]: "filter_mset P (M #∩ N) = filter_mset P M #∩ filter_mset P N"
by (rule multiset_eqI) simp

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_le_exists_conv]
obtain C where B: "B = A + C" by auto
show ?thesis unfolding B by auto
qed

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"

subsubsection ‹Set of elements›

definition set_mset :: "'a multiset ⇒ 'a set"
where "set_mset M = {x. x ∈# M}"

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

lemma set_mset_single [simp]: "set_mset {#b#} = {b}"

lemma set_mset_union [simp]: "set_mset (M + N) = set_mset M ∪ set_mset N"

lemma set_mset_eq_empty_iff [simp]: "(set_mset M = {}) = (M = {#})"
by (auto simp add: set_mset_def multiset_eq_iff)

lemma mem_set_mset_iff [simp]: "(x ∈ set_mset M) = (x ∈# M)"

lemma set_mset_filter [simp]: "set_mset {# x∈#M. P x #} = set_mset M ∩ {x. P x}"

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

lemma finite_Collect_mem [iff]: "finite {x. x ∈# M}"
unfolding set_mset_def[symmetric] by simp

lemma set_mset_mono: "A ≤# B ⟹ set_mset A ⊆ set_mset B"
by (metis mset_leD subsetI mem_set_mset_iff)

lemma ball_set_mset_iff: "(∀x ∈ set_mset M. P x) ⟷ (∀x. x ∈# M ⟶ P x)"
by auto

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"

definition size_multiset :: "('a ⇒ nat) ⇒ 'a multiset ⇒ nat" where
"size_multiset f M = setsum (wcount f M) (set_mset M)"

lemmas size_multiset_eq = size_multiset_def[unfolded wcount_def]

instantiation multiset :: (type) size
begin

definition size_multiset where
size_multiset_overloaded_def: "size_multiset = Multiset.size_multiset (λ_. 0)"
instance ..

end

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

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

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

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

lemma setsum_wcount_Int:
"finite A ⟹ setsum (wcount f N) (A ∩ set_mset N) = setsum (wcount f N) A"
apply (induct rule: finite_induct)
apply simp
apply (simp add: Int_insert_left set_mset_def wcount_def)
done

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

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 multiset_eq_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 setsum_SucD)
apply auto
done

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

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

lemma size_filter_mset_lesseq[simp]: "size (filter_mset f M) ≤ 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 {#}"
assumes add: "⋀M x. P M ⟹ P (M + {#x#})"
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 = N + {#x#}"
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 = A + {#a#}"
by (induct M) auto

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

lemma multi_drop_mem_not_eq: "c ∈# 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_less_size: "(A::'a multiset) <# B ⟹ size A < size B"
proof (induct A arbitrary: B)
case (empty M)
then have "M ≠ {#}" by (simp add: mset_less_empty_nonempty)
then obtain M' x where "M = M' + {#x#}"
by (blast dest: multi_nonempty_split)
then show ?case by simp
next
have IH: "⋀B. S <# B ⟹ size S < size B" by fact
have SxsubT: "S + {#x#} <# T" by fact
then have "x ∈# T" and "S <# T" by (auto dest: mset_less_insertD)
then obtain T' where T: "T = T' + {#x#}"
by (blast dest: multi_member_split)
then have "S <# T'" using SxsubT
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_less_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_less_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_less_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 (F + {#a#})"
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: "F + {#x#} ≤# A"
show "P (F + {#x#})"
proof (rule insert)
from i show "x ∈# A" by (auto dest: mset_le_insertD)
from i have "F ≤# A" by (auto dest: mset_le_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_insert: "fold_mset f s (M + {#x#}) = 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 (M + {#x#}) y"
by (fact comp_fun_commute_funpow)
show ?thesis
proof (cases "x ∈ set_mset M")
case False
then have *: "count (M + {#x#}) x = 1" by simp
from False have "Finite_Set.fold (λy. f y ^^ count (M + {#x#}) 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
by (simp add: fold_mset_def del: count_union)
next
case True
def 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 (M + {#x#}) 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)
with * show ?thesis by (simp add: fold_mset_def del: count_union) simp
qed
qed

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

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

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

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

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 (plus ∘ single ∘ f) {#}"

lemma comp_fun_commute_mset_image: "comp_fun_commute (plus ∘ single ∘ f)"
proof

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

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

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

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

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

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

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

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 mem_set_mset_iff 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) = mset x + {# a #}"

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

lemma mem_set_multiset_eq: "x ∈ set xs = (x ∈# mset xs)"
by (induct xs) auto

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

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

lemma mset_filter: "mset (filter P xs) = {#x ∈# 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 set_count_greater_0: "set x = {a. count (mset x) a > 0}"
by (induct x) auto

lemma distinct_count_atmost_1:
"distinct x = (∀a. count (mset x) a = (if a ∈ set x then 1 else 0))"
apply (induct x, simp, rule iffI, simp_all)
subgoal for a b
apply (rule conjI)
apply (simp_all add: set_mset_mset [symmetric] del: set_mset_mset)
apply (erule_tac x = a in allE, simp)
apply clarify
apply (erule_tac x = aa in allE, simp)
done
done

lemma mset_eq_setD: "mset xs = mset ys ⟹ set xs = set ys"
by (rule) (auto simp add:multiset_eq_iff set_count_greater_0)

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 simp: ac_simps)

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) = mset xs + {#x#}"
by (induct xs) (simp_all add: ac_simps)

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

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

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

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

end

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

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

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

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 add: ac_simps)

subsection ‹Replicate operation›

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

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

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

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

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

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

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

lemma count_le_replicate_mset_le: "n ≤ count M x ⟷ replicate_mset n x ≤# M"
by (auto simp add: assms mset_less_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

subsection ‹Big operators›

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

locale comm_monoid_mset = comm_monoid
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

end

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

declare comp_fun_commute.fold_mset_insert[OF comp_fun_commute_plus_mset, simp]

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

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

begin

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

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

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

end

lemma msetsum_diff:
fixes M N :: "('a :: ordered_cancel_comm_monoid_diff) multiset"
shows "N ≤# M ⟹ msetsum (M - N) = msetsum M - msetsum N"

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

abbreviation Union_mset :: "'a multiset multiset ⇒ 'a multiset"  ("⋃#_" [900] 900)
where "⋃# MM ≡ msetsum MM"

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

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

context comm_monoid_mult
begin

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

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

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

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

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

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

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

end

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

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

subsection ‹Alternative representations›

subsubsection ‹Lists›

context linorder
begin

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

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

text ‹
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 ∈ set ys"
by (auto simp add: mem_set_multiset_eq intro!: ccontr)
ultimately show ?case using Cons.prems by (simp add: insort_key_remove1)
qed

lemma properties_for_sort:
assumes multiset: "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_le: "mset (remdups xs) ≤# mset xs"
by (induct xs) (auto intro: subset_mset.order_trans)

lemma mset_update:
"i < length ls ⟹ mset (ls[i := v]) = mset ls - {#ls ! i#} + {#v#}"
proof (induct ls arbitrary: i)
case Nil then show ?case by simp
next
case (Cons x xs)
show ?case
proof (cases i)
case 0 then show ?thesis by simp
next
case (Suc i')
with Cons show ?thesis
apply simp
apply (subst add.commute [of "{#v#}" "{#x#}"])
apply simp
apply (rule mset_le_multiset_union_diff_commute)
done
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 = M0 + {#a#} ∧ 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 not_less_empty [iff]: "(M, {#}) ∉ mult1 r"

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

lemma all_accessible:
assumes "wf r"
shows "∀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. M + {#b#} ∈ ?W)"
and acc_hyp: "∀M. (M, M0) ∈ ?R ⟶ M + {#a#} ∈ ?W"
have "M0 + {#a#} ∈ ?W"
proof (rule accI [of "M0 + {#a#}"])
fix N
assume "(N, M0 + {#a#}) ∈ ?R"
then consider M where "(M, M0) ∈ ?R" "N = M + {#a#}"
| 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 ⟶ M + {#a#} ∈ ?W" ..
from this and ‹(M, M0) ∈ ?R› have "M + {#a#} ∈ ?W" ..
then show "N ∈ ?W" by (simp only: ‹N = M + {#a#}›)
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. M + {#x#} ∈ ?W" by blast
moreover from add have "M0 + K ∈ ?W" by simp
ultimately have "(M0 + K) + {#x#} ∈ ?W" ..
then show "M0 + (K + {#x#}) ∈ ?W" by (simp only: add.assoc)
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. M + {#a#} ∈ ?W"
proof induct
fix a
assume r: "⋀b. (b, a) ∈ r ⟹ (∀M ∈ ?W. M + {#b#} ∈ ?W)"
show "∀M ∈ ?W. M + {#a#} ∈ ?W"
proof
fix M assume "M ∈ ?W"
then show "M + {#a#} ∈ ?W"
by (rule acc_induct) (rule tedious_reasoning [OF _ r])
qed
qed
from this and ‹M ∈ ?W› show "M + {#a#} ∈ ?W" ..
qed
qed

theorem wf_mult1: "wf r ⟹ wf (mult1 r)"
by (rule acc_wfI) (rule all_accessible)

theorem wf_mult: "wf r ⟹ wf (mult r)"
unfolding mult_def by (rule wf_trancl) (rule wf_mult1)

subsubsection ‹Closure-free presentation›

text ‹One direction.›

lemma mult_implies_one_step:
"trans r ⟹ (M, N) ∈ mult r ⟹
∃I J K. N = I + J ∧ M = I + K ∧ J ≠ {#} ∧
(∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ r)"
apply (unfold mult_def mult1_def set_mset_def)
apply (erule converse_trancl_induct, clarify)
apply (rule_tac x = M0 in exI, simp, clarify)
apply (case_tac "a ∈# K")
apply (rule_tac x = I in exI)
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (drule_tac f = "λM. M - {#a#}" and x="S + T" for S T in arg_cong)
apply blast
apply (subgoal_tac "a ∈# I")
apply (rule_tac x = "I - {#a#}" in exI)
apply (rule_tac x = "J + {#a#}" in exI)
apply (rule_tac x = "K + Ka" in exI)
apply (rule conjI)
apply (simp add: multiset_eq_iff split: nat_diff_split)
apply (rule conjI)
apply (drule_tac f = "λM. M - {#a#}" and x="S + T" for S T in arg_cong, simp)
apply (simp add: multiset_eq_iff split: nat_diff_split)
apply blast
apply (subgoal_tac "a ∈# (M0 + {#a#})")
apply simp
apply (simp (no_asm))
done

lemma one_step_implies_mult_aux:
"∀I J K. size J = n ∧ J ≠ {#} ∧ (∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ r)
⟶ (I + K, I + J) ∈ mult r"
apply (induct n)
apply auto
apply (frule size_eq_Suc_imp_eq_union, clarify)
apply (rename_tac "J'", simp)
apply (erule notE, auto)
apply (case_tac "J' = {#}")
apply (rule r_into_trancl)
apply (simp add: mult1_def set_mset_def, blast)
txt ‹Now we know @{term "J' ≠ {#}"}.›
apply (cut_tac M = K and P = "λx. (x, a) ∈ r" in multiset_partition)
apply (erule_tac P = "∀k ∈ set_mset K. P k" for P in rev_mp)
apply (erule ssubst)
apply (subgoal_tac
"((I + {# x ∈# K. (x, a) ∈ r #}) + {# x ∈# K. (x, a) ∉ r #},
(I + {# x ∈# K. (x, a) ∈ r #}) + J') ∈ mult r")
prefer 2
apply force
apply (erule trancl_trans)
apply (rule r_into_trancl)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
done

lemma one_step_implies_mult:
"trans r ⟹ J ≠ {#} ⟹ ∀k ∈ set_mset K. ∃j ∈ set_mset J. (k, j) ∈ r
⟹ (I + K, I + J) ∈ mult r"
using one_step_implies_mult_aux by blast

subsubsection ‹Partial-order properties›

definition less_multiset :: "'a::order multiset ⇒ 'a multiset ⇒ bool"  (infix "#⊂#" 50)
where "M' #⊂# M ⟷ (M', M) ∈ mult {(x', x). x' < x}"

definition le_multiset :: "'a::order multiset ⇒ 'a multiset ⇒ bool"  (infix "#⊆#" 50)
where "M' #⊆# M ⟷ M' #⊂# M ∨ M' = M"

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

interpretation multiset_order: order le_multiset less_multiset
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 (rule transI) simp
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 "class.order (le_multiset :: 'a multiset ⇒ _) less_multiset"
by standard (auto simp add: le_multiset_def irrefl dest: trans)
qed

lemma mult_less_irrefl [elim!]:
fixes M :: "'a::order 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"
apply (unfold mult1_def)
apply auto
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
done

lemma union_less_mono2: "B #⊂# D ⟹ C + B #⊂# C + (D::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
apply (erule trancl_induct)
apply (blast intro: mult1_union)
apply (blast intro: mult1_union trancl_trans)
done

lemma union_less_mono1: "B #⊂# D ⟹ B + C #⊂# D + (C::'a::order multiset)"
apply (subst add.commute [of B C])
apply (subst add.commute [of D C])
apply (erule union_less_mono2)
done

lemma union_less_mono:
fixes A B C D :: "'a::order multiset"
shows "A #⊂# C ⟹ B #⊂# D ⟹ A + B #⊂# C + D"
by (blast intro!: union_less_mono1 union_less_mono2 multiset_order.less_trans)

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

subsubsection ‹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 simp: ac_simps)
thus ?thesis by blast
next
case 2
let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
have "{#x#} + X = ?A' + Z"
"{#y#} + Y = ?B' + Z"
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 single}, T --> msetT T) \$ x
| mk_mset T (x :: xs) =
Const (@{const_name plus}, msetT T --> msetT T --> msetT T) \$
mk_mset T [x] \$ mk_mset T xs

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

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

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

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

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

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

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_less_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 mult_less_not_refl: "¬ M #⊂# (M::'a::order multiset)"
by (fact multiset_order.less_irrefl)

lemma mult_less_trans: "K #⊂# M ⟹ M #⊂# N ⟹ K #⊂# (N::'a::order multiset)"
by (fact multiset_order.less_trans)

lemma mult_less_not_sym: "M #⊂# N ⟹ ¬ N #⊂# (M::'a::order multiset)"
by (fact multiset_order.less_not_sym)

lemma mult_less_asym: "M #⊂# N ⟹ (¬ P ⟹ N #⊂# (M::'a::order multiset)) ⟹ P"
by (fact multiset_order.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 (t1, t2) =>
Const (@{const_name plus_class.plus}, T --> T --> T) \$ t1 \$ t2)
(map (curry (op \$) (Const (@{const_name single}, elem_T --> T))) ts))
end
| multiset_postproc _ _ _ _ t = t
in Nitpick_Model.register_term_postprocessor @{typ "'a multiset"} multiset_postproc end
›

subsection ‹Naive implementation using lists›

code_datatype mset

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

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

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

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

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)
done

declare size_mset [code]

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

lemma ms_lesseq_impl: "(ms_lesseq_impl xs ys = None ⟷ ¬ mset xs ≤# mset ys) ∧
(ms_lesseq_impl xs ys = Some True ⟷ mset xs <# mset ys) ∧
(ms_lesseq_impl xs ys = Some False ⟶ mset xs = mset ys)"
proof (induct xs arbitrary: ys)
case (Nil ys)
show ?case by (auto simp: mset_less_empty_nonempty)
next
case (Cons x xs ys)
show ?case
proof (cases "List.extract (op = x) ys")
case None
hence x: "x ∉ 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 = mset (ys1 @ ys2) + {#x#}"
by (auto simp: ac_simps)
show ?thesis unfolding ms_lesseq_impl.simps
unfolding Some option.simps split
unfolding id
using Cons[of "ys1 @ ys2"]
unfolding subset_mset_def subseteq_mset_def by auto
qed
qed

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

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

instantiation multiset :: (equal) equal
begin

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

instance

end

lemma [code]: "msetsum (mset xs) = listsum xs"

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

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

lemma ex_mset_zip_left:
assumes "length xs = length ys" "mset xs' = mset xs"
shows "∃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)

def xsa ≡ "take j xs' @ drop (Suc j) xs'"
have "mset xs' = {#x#} + mset xsa"
unfolding xsa_def using j_len nth_j
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

def 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' add.commute)
ultimately show ?case
by blast
qed

lemma list_all2_reorder_left_invariance:
assumes rel: "list_all2 R xs ys" and ms_x: "mset xs' = mset xs"
shows "∃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))

bnf "'a multiset"
map: image_mset
sets: set_mset
bd: natLeq
wits: "{#}"
rel: rel_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 (no_asm),
metis One_nat_def Un_iff count_single mem_set_mset_iff set_mset_union zero_less_Suc)
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 =
(BNF_Def.Grp {x. set_mset x ⊆ {(x, y). R x y}} (image_mset fst))¯¯ OO
BNF_Def.Grp {x. set_mset x ⊆ {(x, y). R x y}} (image_mset snd)" for R
unfolding rel_mset_def[abs_def] BNF_Def.Grp_def OO_def
apply (rule ext)+
apply auto
apply (rule_tac x = "mset (zip xs ys)" in exI; auto)
apply (metis list_all2_lengthD map_fst_zip mset_map)
apply (auto simp: list_all2_iff)[1]
apply (metis list_all2_lengthD map_snd_zip mset_map)
apply (auto simp: list_all2_iff)[1]
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
qed

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

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

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

lemma rel_mset_Plus:
assumes ab: "R a b"
and MN: "rel_mset R M N"
shows "rel_mset R (M + {#a#}) (N + {#b#})"
proof -
have "∃ya. image_mset fst y + {#a#} = image_mset fst ya ∧
image_mset snd y + {#b#} = 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 _ "y + {#(a,b)#}"]) auto
thus ?thesis
using assms
unfolding multiset.rel_compp_Grp Grp_def by blast
qed

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

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

lemma msed_map_invR:
assumes "image_mset f M = N + {#b#}"
shows "∃M1 a. M = M1 + {#a#} ∧ 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 mem_set_mset_iff union_single_eq_member)
then obtain M1 where M: "M = M1 + {#a#}" using multi_member_split by metis
have "image_mset f M1 = N" using assms unfolding M fa[symmetric] by simp
thus ?thesis using M fa by blast
qed

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

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

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

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

text ‹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}