Theory Multiset

Up to index of Isabelle/HOL/Imperative_HOL

theory Multiset
imports Main

(*  Title:      HOL/Library/Multiset.thy
Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
*)


header {* (Finite) multisets *}

theory Multiset
imports Main
begin


subsection {* The type of multisets *}

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

proof
show "(λx. 0::nat) ∈ ?multiset" by simp
qed

lemmas multiset_typedef = Abs_multiset_inverse count_inverse count

abbreviation Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where
"a :# M == 0 < count M a"


notation (xsymbols)
Melem (infix "∈#" 50)


lemma multiset_ext_iff:
"M = N <-> (∀a. count M a = count N a)"

by (simp only: count_inject [symmetric] expand_fun_eq)

lemma multiset_ext:
"(!!x. count A x = count B x) ==> A = B"

using multiset_ext_iff by auto

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


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

by (simp add: multiset_def)

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

by (simp add: multiset_def)

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

by (simp add: multiset_def)

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 MCollect_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 MCollect_preserves_multiset



subsection {* Representing multisets *}

text {* Multiset comprehension *}

definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
"MCollect M P = Abs_multiset (λx. if P x then count M x else 0)"


syntax
"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})")

translations
"{#x :# M. P#}" == "CONST MCollect M (λx. P)"



text {* Multiset enumeration *}

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


definition Mempty_def:
"0 = Abs_multiset (λa. 0)"


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


definition union_def:
"M + N = Abs_multiset (λa. count M a + count N a)"


instance ..

end

definition single :: "'a => 'a multiset" where
"single a = Abs_multiset (λb. if b = a then 1 else 0)"


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

translations
"{#x, xs#}" == "{#x#} + {#xs#}"
"{#x#}" == "CONST single x"


lemma count_empty [simp]: "count {#} a = 0"
by (simp add: Mempty_def in_multiset multiset_typedef)

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


subsection {* Basic operations *}

subsubsection {* Union *}

lemma count_union [simp]: "count (M + N) a = count M a + count N a"
by (simp add: union_def in_multiset multiset_typedef)

instance multiset :: (type) cancel_comm_monoid_add proof
qed (simp_all add: multiset_ext_iff)


subsubsection {* Difference *}

instantiation multiset :: (type) minus
begin


definition diff_def:
"M - N = Abs_multiset (λa. count M a - count N a)"


instance ..

end

lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
by (simp add: diff_def in_multiset multiset_typedef)

lemma diff_empty [simp]: "M - {#} = M ∧ {#} - M = {#}"
by(simp add: multiset_ext_iff)

lemma diff_cancel[simp]: "A - A = {#}"
by (rule multiset_ext) simp

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

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

lemma insert_DiffM:
"x ∈# M ==> {#x#} + (M - {#x#}) = M"

by (clarsimp simp: multiset_ext_iff)

lemma insert_DiffM2 [simp]:
"x ∈# M ==> M - {#x#} + {#x#} = M"

by (clarsimp simp: multiset_ext_iff)

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

by (auto simp add: multiset_ext_iff)

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

by (simp add: multiset_ext_iff)

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

by (auto simp add: multiset_ext_iff)

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

by (simp add: multiset_ext_iff)


subsubsection {* Equality of multisets *}

lemma single_not_empty [simp]: "{#a#} ≠ {#} ∧ {#} ≠ {#a#}"
by (simp add: multiset_ext_iff)

lemma single_eq_single [simp]: "{#a#} = {#b#} <-> a = b"
by (auto simp add: multiset_ext_iff)

lemma union_eq_empty [iff]: "M + N = {#} <-> M = {#} ∧ N = {#}"
by (auto simp add: multiset_ext_iff)

lemma empty_eq_union [iff]: "{#} = M + N <-> M = {#} ∧ N = {#}"
by (auto simp add: multiset_ext_iff)

lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} <-> False"
by (auto simp add: multiset_ext_iff)

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

by (auto simp add: multiset_ext_iff)

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
assume ?rhs then show ?lhs by auto
next
assume ?lhs thus ?rhs
by(simp add: multiset_ext_iff split:if_splits) (metis add_is_1)
qed

lemma single_is_union:
"{#a#} = M + N <-> {#a#} = M ∧ N = {#} ∨ M = {#} ∧ {#a#} = N"

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

lemma add_eq_conv_diff:
"M + {#a#} = N + {#b#} <-> M = N ∧ a = b ∨ M = N - {#a#} + {#b#} ∧ N = M - {#b#} + {#a#}" (is "?lhs = ?rhs")

(* shorter: by (simp add: multiset_ext_iff) fastsimp *)
proof
assume ?rhs then show ?lhs
by (auto simp add: add_assoc add_commute [of "{#b#}"])
(drule sym, simp add: add_assoc [symmetric])

next
assume ?lhs
show ?rhs
proof (cases "a = b")
case True with `?lhs` show ?thesis by simp
next
case False
from `?lhs` have "a ∈# 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

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

by (auto simp add: add_eq_conv_diff)


subsubsection {* Pointwise ordering induced by count *}

instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le
begin


definition less_eq_multiset :: "'a multiset => 'a multiset => bool" where
mset_le_def: "A ≤ B <-> (∀a. count A a ≤ count B a)"


definition less_multiset :: "'a multiset => 'a multiset => bool" where
mset_less_def: "(A::'a multiset) < B <-> A ≤ B ∧ A ≠ B"


instance proof
qed (auto simp add: mset_le_def mset_less_def multiset_ext_iff intro: order_trans antisym)

end

lemma mset_less_eqI:
"(!!x. count A x ≤ count B x) ==> A ≤ B"

by (simp add: mset_le_def)

lemma mset_le_exists_conv:
"(A::'a multiset) ≤ B <-> (∃C. B = A + C)"

apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
apply (auto intro: multiset_ext_iff [THEN iffD2])
done

lemma mset_le_mono_add_right_cancel [simp]:
"(A::'a multiset) + C ≤ B + C <-> A ≤ B"

by (fact add_le_cancel_right)

lemma mset_le_mono_add_left_cancel [simp]:
"C + (A::'a multiset) ≤ C + B <-> A ≤ B"

by (fact add_le_cancel_left)

lemma mset_le_mono_add:
"(A::'a multiset) ≤ B ==> C ≤ D ==> A + C ≤ B + D"

by (fact add_mono)

lemma mset_le_add_left [simp]:
"(A::'a multiset) ≤ A + B"

unfolding mset_le_def by auto

lemma mset_le_add_right [simp]:
"B ≤ (A::'a multiset) + B"

unfolding mset_le_def by auto

lemma mset_le_single:
"a :# B ==> {#a#} ≤ B"

by (simp add: mset_le_def)

lemma multiset_diff_union_assoc:
"C ≤ B ==> (A::'a multiset) + B - C = A + (B - C)"

by (simp add: multiset_ext_iff mset_le_def)

lemma mset_le_multiset_union_diff_commute:
"B ≤ A ==> (A::'a multiset) - B + C = A + C - B"

by (simp add: multiset_ext_iff mset_le_def)

lemma mset_lessD: "A < B ==> x ∈# A ==> x ∈# B"
apply (clarsimp simp: mset_le_def mset_less_def)
apply (erule_tac x=x in allE)
apply auto
done

lemma mset_leD: "A ≤ B ==> x ∈# A ==> x ∈# B"
apply (clarsimp simp: mset_le_def mset_less_def)
apply (erule_tac x = x in allE)
apply auto
done

lemma mset_less_insertD: "(A + {#x#} < B) ==> (x ∈# B ∧ A < B)"
apply (rule conjI)
apply (simp add: mset_lessD)
apply (clarsimp simp: mset_le_def mset_less_def)
apply safe
apply (erule_tac x = a in allE)
apply (auto split: split_if_asm)
done

lemma mset_le_insertD: "(A + {#x#} ≤ B) ==> (x ∈# B ∧ A ≤ B)"
apply (rule conjI)
apply (simp add: mset_leD)
apply (force simp: mset_le_def mset_less_def split: split_if_asm)
done

lemma mset_less_of_empty[simp]: "A < {#} <-> False"
by (auto simp add: mset_less_def mset_le_def multiset_ext_iff)

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

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

lemma mset_less_add_bothsides:
"T + {#x#} < S + {#x#} ==> T < S"

by (fact add_less_imp_less_right)

lemma mset_less_empty_nonempty:
"{#} < S <-> S ≠ {#}"

by (auto simp: mset_le_def mset_less_def)

lemma mset_less_diff_self:
"c ∈# B ==> B - {#c#} < B"

by (auto simp: mset_le_def mset_less_def multiset_ext_iff)


subsubsection {* Intersection *}

instantiation multiset :: (type) semilattice_inf
begin


definition inf_multiset :: "'a multiset => 'a multiset => 'a multiset" where
multiset_inter_def: "inf_multiset A B = A - (A - B)"


instance proof -
have aux: "!!m n q :: nat. m ≤ n ==> m ≤ q ==> m ≤ n - (n - q)" by arith
show "OFCLASS('a multiset, semilattice_inf_class)" proof
qed (auto simp add: multiset_inter_def mset_le_def aux)
qed

end

abbreviation multiset_inter :: "'a multiset => 'a multiset => 'a multiset" (infixl "#∩" 70) where
"multiset_inter ≡ inf"


lemma multiset_inter_count:
"count (A #∩ B) x = min (count A x) (count B x)"

by (simp add: multiset_inter_def multiset_typedef)

lemma multiset_inter_single: "a ≠ b ==> {#a#} #∩ {#b#} = {#}"
by (rule multiset_ext) (auto simp add: multiset_inter_count)

lemma multiset_union_diff_commute:
assumes "B #∩ C = {#}"
shows "A + B - C = A - C + B"

proof (rule multiset_ext)
fix x
from assms have "min (count B x) (count C x) = 0"
by (auto simp add: multiset_inter_count multiset_ext_iff)
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


subsubsection {* Comprehension (filter) *}

lemma count_MCollect [simp]:
"count {# x:#M. P x #} a = (if P a then count M a else 0)"

by (simp add: MCollect_def in_multiset multiset_typedef)

lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
by (rule multiset_ext) simp

lemma MCollect_single [simp]:
"MCollect {#x#} P = (if P x then {#x#} else {#})"

by (rule multiset_ext) simp

lemma MCollect_union [simp]:
"MCollect (M + N) f = MCollect M f + MCollect N f"

by (rule multiset_ext) simp


subsubsection {* Set of elements *}

definition set_of :: "'a multiset => 'a set" where
"set_of M = {x. x :# M}"


lemma set_of_empty [simp]: "set_of {#} = {}"
by (simp add: set_of_def)

lemma set_of_single [simp]: "set_of {#b#} = {b}"
by (simp add: set_of_def)

lemma set_of_union [simp]: "set_of (M + N) = set_of M ∪ set_of N"
by (auto simp add: set_of_def)

lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
by (auto simp add: set_of_def multiset_ext_iff)

lemma mem_set_of_iff [simp]: "(x ∈ set_of M) = (x :# M)"
by (auto simp add: set_of_def)

lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M ∩ {x. P x}"
by (auto simp add: set_of_def)

lemma finite_set_of [iff]: "finite (set_of M)"
using count [of M] by (simp add: multiset_def set_of_def)


subsubsection {* Size *}

instantiation multiset :: (type) size
begin


definition size_def:
"size M = setsum (count M) (set_of M)"


instance ..

end

lemma size_empty [simp]: "size {#} = 0"
by (simp add: size_def)

lemma size_single [simp]: "size {#b#} = 1"
by (simp add: size_def)

lemma setsum_count_Int:
"finite A ==> setsum (count N) (A ∩ set_of N) = setsum (count N) A"

apply (induct rule: finite_induct)
apply simp
apply (simp add: Int_insert_left set_of_def)
done

lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
apply (unfold size_def)
apply (subgoal_tac "count (M + N) = (λa. count M a + count N a)")
prefer 2
apply (rule ext, simp)
apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
apply (subst Int_commute)
apply (simp (no_asm_simp) add: setsum_count_Int)
done

lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
by (auto simp add: size_def multiset_ext_iff)

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 (unfold size_def)
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


subsection {* Induction and case splits *}

lemma setsum_decr:
"finite F ==> (0::nat) < f a ==>
setsum (f (a := f a - 1)) F = (if a∈F then setsum f F - 1 else setsum f F)"

apply (induct rule: finite_induct)
apply auto
apply (drule_tac a = a in mk_disjoint_insert, auto)
done

lemma rep_multiset_induct_aux:
assumes 1: "P (λa. (0::nat))"
and 2: "!!f b. f ∈ multiset ==> P f ==> P (f (b := f b + 1))"
shows "∀f. f ∈ multiset --> setsum f {x. f x ≠ 0} = n --> P f"

apply (unfold multiset_def)
apply (induct_tac n, simp, clarify)
apply (subgoal_tac "f = (λa.0)")
apply simp
apply (rule 1)
apply (rule ext, force, clarify)
apply (frule setsum_SucD, clarify)
apply (rename_tac a)
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
prefer 2
apply (rule finite_subset)
prefer 2
apply assumption
apply simp
apply blast
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
prefer 2
apply (rule ext)
apply (simp (no_asm_simp))
apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
apply (erule allE, erule impE, erule_tac [2] mp, blast)
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
apply (subgoal_tac "{x. x ≠ a --> f x ≠ 0} = {x. f x ≠ 0}")
prefer 2
apply blast
apply (subgoal_tac "{x. x ≠ a ∧ f x ≠ 0} = {x. f x ≠ 0} - {a}")
prefer 2
apply blast
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
done

theorem rep_multiset_induct:
"f ∈ multiset ==> P (λa. 0) ==>
(!!f b. f ∈ multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"

using rep_multiset_induct_aux by blast

theorem multiset_induct [case_names empty add, induct type: multiset]:
assumes empty: "P {#}"
and add: "!!M x. P M ==> P (M + {#x#})"
shows "P M"

proof -
note defns = union_def single_def Mempty_def
note add' = add [unfolded defns, simplified]
have aux: "!!a::'a. count (Abs_multiset (λb. if b = a then 1 else 0)) =
(λb. if b = a then 1 else 0)"
by (simp add: Abs_multiset_inverse in_multiset)
show ?thesis
apply (rule count_inverse [THEN subst])
apply (rule count [THEN rep_multiset_induct])
apply (rule empty [unfolded defns])
apply (subgoal_tac "f(b := f b + 1) = (λa. f a + (if a=b then 1 else 0))")
prefer 2
apply (simp add: expand_fun_eq)
apply (erule ssubst)
apply (erule Abs_multiset_inverse [THEN subst])
apply (drule add')
apply (simp add: aux)
done
qed

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

lemma multiset_cases [cases type, case_names empty add]:
assumes em: "M = {#} ==> P"
assumes add: "!!N x. M = N + {#x#} ==> P"
shows "P"

proof (cases "M = {#}")
assume "M = {#}" then show ?thesis using em by simp
next
assume "M ≠ {#}"
then obtain M' m where "M = M' + {#m#}"
by (blast dest: multi_nonempty_split)
then show ?thesis using add by simp
qed

lemma multi_member_split: "x ∈# M ==> ∃A. M = A + {#x#}"
apply (cases M)
apply simp
apply (rule_tac x="M - {#x#}" in exI, simp)
done

lemma multi_drop_mem_not_eq: "c ∈# 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_ext_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
case (add S x T)
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
by (blast intro: mset_less_add_bothsides)
then have "size S < size T'" using IH by simp
then show ?case using T by simp
qed


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

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

text {* proper multiset subset *}

definition
mset_less_rel :: "('a multiset * 'a multiset) set" where
"mset_less_rel = {(A,B). A < B}"


lemma multiset_add_sub_el_shuffle:
assumes "c ∈# 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#}"
by (simp add: add_ac)
then show ?thesis using B by simp
qed

lemma wf_mset_less_rel: "wf mset_less_rel"
apply (unfold mset_less_rel_def)
apply (rule wf_measure [THEN wf_subset, where f1=size])
apply (clarsimp simp: measure_def inv_image_def mset_less_size)
done

text {* The induction rules: *}

lemma full_multiset_induct [case_names less]:
assumes ih: "!!B. ∀(A::'a multiset). A < B --> P A ==> P B"
shows "P B"

apply (rule wf_mset_less_rel [THEN wf_induct])
apply (rule ih, auto simp: mset_less_rel_def)
done

lemma multi_subset_induct [consumes 2, case_names empty add]:
assumes "F ≤ 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 {* Alternative representations *}

subsubsection {* Lists *}

primrec multiset_of :: "'a list => 'a multiset" where
"multiset_of [] = {#}" |
"multiset_of (a # x) = multiset_of x + {# a #}"


lemma in_multiset_in_set:
"x ∈# multiset_of xs <-> x ∈ set xs"

by (induct xs) simp_all

lemma count_multiset_of:
"count (multiset_of xs) x = length (filter (λy. x = y) xs)"

by (induct xs) simp_all

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

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

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

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

lemma multiset_of_append [simp]:
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"

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

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

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

lemma distinct_count_atmost_1:
"distinct x = (! a. count (multiset_of x) a = (if a ∈ set x then 1 else 0))"

apply (induct x, simp, rule iffI, simp_all)
apply (rule conjI)
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
apply (erule_tac x = a in allE, simp, clarify)
apply (erule_tac x = aa in allE, simp)
done

lemma multiset_of_eq_setD:
"multiset_of xs = multiset_of ys ==> set xs = set ys"

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

lemma set_eq_iff_multiset_of_eq_distinct:
"distinct x ==> distinct y ==>
(set x = set y) = (multiset_of x = multiset_of y)"

by (auto simp: multiset_ext_iff distinct_count_atmost_1)

lemma set_eq_iff_multiset_of_remdups_eq:
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"

apply (rule iffI)
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
apply (drule distinct_remdups [THEN distinct_remdups
[THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])

apply simp
done

lemma multiset_of_compl_union [simp]:
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. ¬P x] = multiset_of xs"

by (induct xs) (auto simp: add_ac)

lemma count_filter:
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"

by (induct xs) auto

lemma nth_mem_multiset_of: "i < length ls ==> (ls ! i) :# multiset_of ls"
apply (induct ls arbitrary: i)
apply simp
apply (case_tac i)
apply auto
done

lemma multiset_of_remove1[simp]:
"multiset_of (remove1 a xs) = multiset_of xs - {#a#}"

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

lemma multiset_of_eq_length:
assumes "multiset_of xs = multiset_of ys"
shows "length xs = length ys"

using assms proof (induct xs arbitrary: ys)
case Nil then show ?case by simp
next
case (Cons x xs)
then have "x ∈# multiset_of ys" by (simp add: union_single_eq_member)
then have "x ∈ set ys" by (simp add: in_multiset_in_set)
from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)"
by simp
with Cons.hyps have "length xs = length (remove1 x ys)" .
with `x ∈ set ys` show ?case
by (auto simp add: length_remove1 dest: length_pos_if_in_set)
qed

lemma (in linorder) multiset_of_insort [simp]:
"multiset_of (insort x xs) = {#x#} + multiset_of xs"

by (induct xs) (simp_all add: ac_simps)

lemma (in linorder) multiset_of_sort [simp]:
"multiset_of (sort xs) = multiset_of xs"

by (induct xs) (simp_all add: ac_simps)

text {*
This lemma shows which properties suffice to show that a function
@{text "f"} with @{text "f xs = ys"} behaves like sort.
*}


lemma (in linorder) properties_for_sort:
"multiset_of ys = multiset_of xs ==> sorted ys ==> sort xs = ys"

proof (induct xs arbitrary: ys)
case Nil then show ?case by simp
next
case (Cons x xs)
then have "x ∈ set ys"
by (auto simp add: mem_set_multiset_eq intro!: ccontr)
with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
qed

lemma multiset_of_remdups_le: "multiset_of (remdups xs) ≤ multiset_of xs"
by (induct xs) (auto intro: order_trans)

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

proof (induct ls arbitrary: i)
case Nil then show ?case by simp
next
case (Cons x xs)
show ?case
proof (cases i)
case 0 then show ?thesis by simp
next
case (Suc i')
with Cons show ?thesis
apply simp
apply (subst add_assoc)
apply (subst add_commute [of "{#v#}" "{#x#}"])
apply (subst add_assoc [symmetric])
apply simp
apply (rule mset_le_multiset_union_diff_commute)
apply (simp add: mset_le_single nth_mem_multiset_of)
done
qed
qed

lemma multiset_of_swap:
"i < length ls ==> j < length ls ==>
multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"

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


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

definition count_of :: "('a × nat) list => 'a => nat" where
"count_of xs x = (case map_of xs x of None => 0 | Some n => n)"


lemma count_of_multiset:
"count_of xs ∈ multiset"

proof -
let ?A = "{x::'a. 0 < (case map_of xs x of None => 0::nat | Some (n::nat) => n)}"
have "?A ⊆ dom (map_of xs)"
proof
fix x
assume "x ∈ ?A"
then have "0 < (case map_of xs x of None => 0::nat | Some (n::nat) => n)" by simp
then have "map_of xs x ≠ None" by (cases "map_of xs x") auto
then show "x ∈ dom (map_of xs)" by auto
qed
with finite_dom_map_of [of xs] have "finite ?A"
by (auto intro: finite_subset)
then show ?thesis
by (simp add: count_of_def expand_fun_eq multiset_def)
qed

lemma count_simps [simp]:
"count_of [] = (λ_. 0)"
"count_of ((x, n) # xs) = (λy. if x = y then n else count_of xs y)"

by (simp_all add: count_of_def expand_fun_eq)

lemma count_of_empty:
"x ∉ fst ` set xs ==> count_of xs x = 0"

by (induct xs) (simp_all add: count_of_def)

lemma count_of_filter:
"count_of (filter (P o fst) xs) x = (if P x then count_of xs x else 0)"

by (induct xs) auto

definition Bag :: "('a × nat) list => 'a multiset" where
"Bag xs = Abs_multiset (count_of xs)"


code_datatype Bag

lemma count_Bag [simp, code]:
"count (Bag xs) = count_of xs"

by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)

lemma Mempty_Bag [code]:
"{#} = Bag []"

by (simp add: multiset_ext_iff)

lemma single_Bag [code]:
"{#x#} = Bag [(x, 1)]"

by (simp add: multiset_ext_iff)

lemma MCollect_Bag [code]:
"MCollect (Bag xs) P = Bag (filter (P o fst) xs)"

by (simp add: multiset_ext_iff count_of_filter)

lemma mset_less_eq_Bag [code]:
"Bag xs ≤ A <-> (∀(x, n) ∈ set xs. count_of xs x ≤ count A x)"
(is "?lhs <-> ?rhs")

proof
assume ?lhs then show ?rhs
by (auto simp add: mset_le_def count_Bag)
next
assume ?rhs
show ?lhs
proof (rule mset_less_eqI)
fix x
from `?rhs` have "count_of xs x ≤ count A x"
by (cases "x ∈ fst ` set xs") (auto simp add: count_of_empty)
then show "count (Bag xs) x ≤ count A x"
by (simp add: mset_le_def count_Bag)
qed
qed

instantiation multiset :: (eq) eq
begin


definition
"HOL.eq A B <-> (A::'a multiset) ≤ B ∧ B ≤ A"


instance proof
qed (simp add: eq_multiset_def eq_iff)

end

definition (in term_syntax)
bagify :: "('a::typerep × nat) list × (unit => Code_Evaluation.term)
=> 'a multiset × (unit => Code_Evaluation.term)"
where
[code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {·} xs"


notation fcomp (infixl "o>" 60)
notation scomp (infixl "o->" 60)

instantiation multiset :: (random) random
begin


definition
"Quickcheck.random i = Quickcheck.random i o-> (λxs. Pair (bagify xs))"


instance ..

end

no_notation fcomp (infixl "o>" 60)
no_notation scomp (infixl "o->" 60)

hide_const (open) bagify


subsection {* The multiset order *}

subsubsection {* Well-foundedness *}

definition mult1 :: "('a × 'a) set => ('a multiset × 'a multiset) set" where
[code del]: "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
[code del]: "mult r = (mult1 r)+"


lemma not_less_empty [iff]: "(M, {#}) ∉ mult1 r"
by (simp add: mult1_def)

lemma less_add: "(N, M0 + {#a#}) ∈ mult1 r ==>
(∃M. (M, M0) ∈ mult1 r ∧ N = M + {#a#}) ∨
(∃K. (∀b. b :# K --> (b, a) ∈ r) ∧ N = M0 + K)"

(is "_ ==> ?case1 (mult1 r) ∨ ?case2")

proof (unfold mult1_def)
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"
let ?case1 = "?case1 {(N, M). ?R N M}"

assume "(N, M0 + {#a#}) ∈ {(N, M). ?R N M}"
then have "∃a' M0' K.
M0 + {#a#} = M0' + {#a'#} ∧ N = M0' + K ∧ ?r K a'"
by simp
then show "?case1 ∨ ?case2"
proof (elim exE conjE)
fix a' M0' K
assume N: "N = M0' + K" and r: "?r K a'"
assume "M0 + {#a#} = M0' + {#a'#}"
then have "M0 = M0' ∧ a = a' ∨
(∃K'. M0 = K' + {#a'#} ∧ M0' = K' + {#a#})"

by (simp only: add_eq_conv_ex)
then show ?thesis
proof (elim disjE conjE exE)
assume "M0 = M0'" "a = a'"
with N r have "?r K a ∧ N = M0 + K" by simp
then have ?case2 .. then show ?thesis ..
next
fix K'
assume "M0' = K' + {#a#}"
with N have n: "N = K' + K + {#a#}" by (simp add: add_ac)

assume "M0 = K' + {#a'#}"
with r have "?R (K' + K) M0" by blast
with n have ?case1 by simp then show ?thesis ..
qed
qed
qed

lemma all_accessible: "wf r ==> ∀M. M ∈ acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "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 have "((∃M. (M, M0) ∈ ?R ∧ N = M + {#a#}) ∨
(∃K. (∀b. b :# K --> (b, a) ∈ r) ∧ N = M0 + K))"

by (rule less_add)
then show "N ∈ ?W"
proof (elim exE disjE conjE)
fix M assume "(M, M0) ∈ ?R" and N: "N = M + {#a#}"
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)
next
fix K
assume N: "N = M0 + K"
assume "∀b. b :# K --> (b, a) ∈ r"
then have "M0 + K ∈ ?W"
proof (induct K)
case empty
from M0 show "M0 + {#} ∈ ?W" by simp
next
case (add K x)
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: N)
qed
qed
} note tedious_reasoning = this

assume wf: "wf r"
fix M
show "M ∈ ?W"
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 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_of K. ∃j ∈ set_of J. (k, j) ∈ r)"

apply (unfold mult_def mult1_def set_of_def)
apply (erule converse_trancl_induct, clarify)
apply (rule_tac x = M0 in exI, simp, clarify)
apply (case_tac "a :# K")
apply (rule_tac x = I in exI)
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (simp (no_asm_simp) add: add_assoc [symmetric])
apply (drule_tac f = "λM. M - {#a#}" in arg_cong)
apply (simp add: diff_union_single_conv)
apply (simp (no_asm_use) add: trans_def)
apply blast
apply (subgoal_tac "a :# I")
apply (rule_tac x = "I - {#a#}" in exI)
apply (rule_tac x = "J + {#a#}" in exI)
apply (rule_tac x = "K + Ka" in exI)
apply (rule conjI)
apply (simp add: multiset_ext_iff split: nat_diff_split)
apply (rule conjI)
apply (drule_tac f = "λM. M - {#a#}" in arg_cong, simp)
apply (simp add: multiset_ext_iff split: nat_diff_split)
apply (simp (no_asm_use) add: trans_def)
apply blast
apply (subgoal_tac "a :# (M0 + {#a#})")
apply simp
apply (simp (no_asm))
done

lemma one_step_implies_mult_aux:
"trans r ==>
∀I J K. (size J = n ∧ J ≠ {#} ∧ (∀k ∈ set_of K. ∃j ∈ set_of J. (k, j) ∈ r))
--> (I + K, I + J) ∈ mult r"

apply (induct_tac n, auto)
apply (frule size_eq_Suc_imp_eq_union, clarify)
apply (rename_tac "J'", simp)
apply (erule notE, auto)
apply (case_tac "J' = {#}")
apply (simp add: mult_def)
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def, blast)
txt {* Now we know @{term "J' ≠ {#}"}. *}
apply (cut_tac M = K and P = "λx. (x, a) ∈ r" in multiset_partition)
apply (erule_tac P = "∀k ∈ set_of K. ?P k" in rev_mp)
apply (erule ssubst)
apply (simp add: Ball_def, auto)
apply (subgoal_tac
"((I + {# x :# K. (x, a) ∈ r #}) + {# x :# K. (x, a) ∉ r #},
(I + {# x :# K. (x, a) ∈ r #}) + J') ∈ mult r"
)

prefer 2
apply force
apply (simp (no_asm_use) add: add_assoc [symmetric] mult_def)
apply (erule trancl_trans)
apply (rule r_into_trancl)
apply (simp add: mult1_def set_of_def)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
apply (simp add: add_ac)
done

lemma one_step_implies_mult:
"trans r ==> J ≠ {#} ==> ∀k ∈ set_of K. ∃j ∈ set_of 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 (xsymbols) less_multiset (infix "⊂#" 50)
notation (xsymbols) le_multiset (infix "⊆#" 50)

interpretation multiset_order: order le_multiset less_multiset
proof -
have irrefl: "!!M :: 'a multiset. ¬ M ⊂# M"
proof
fix M :: "'a multiset"
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_of K. ∃j∈set_of 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_of K. ∃j∈set_of J. (k, j) ∈ {(x, y). x < y})"
by blast
then have aux1: "K ≠ {#}" and aux2: "∀k∈set_of K. ∃j∈set_of K. k < j" by auto
have "finite (set_of K)" by simp
moreover note aux2
ultimately have "set_of K = {}"
by (induct rule: finite_induct) (auto intro: order_less_trans)
with aux1 show False by simp
qed
have trans: "!!K M N :: 'a multiset. K ⊂# M ==> M ⊂# N ==> K ⊂# N"
unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
show "class.order (le_multiset :: 'a multiset => _) less_multiset" proof
qed (auto simp add: le_multiset_def irrefl dest: trans)
qed

lemma mult_less_irrefl [elim!]:
"M ⊂# (M::'a::order multiset) ==> R"

by (simp add: multiset_order.less_irrefl)


subsubsection {* Monotonicity of multiset union *}

lemma mult1_union:
"(B, D) ∈ mult1 r ==> trans 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)
apply (simp add: add_assoc)
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 transI order_less_trans r_into_trancl)
apply (blast intro: mult1_union transI order_less_trans r_into_trancl 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:
"A ⊂# C ==> B ⊂# D ==> A + B ⊂# C + (D::'a::order multiset)"

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

interpretation multiset_order: ordered_ab_semigroup_add plus le_multiset less_multiset
proof
qed (auto simp add: le_multiset_def intro: union_less_mono2)


subsection {* The fold combinator *}

text {*
The intended behaviour is
@{text "fold_mset f z {#x1, ..., xn#} = f x1 (… (f xn z)…)"}
if @{text f} is associative-commutative.
*}


text {*
The graph of @{text "fold_mset"}, @{text "z"}: the start element,
@{text "f"}: folding function, @{text "A"}: the multiset, @{text
"y"}: the result.
*}

inductive
fold_msetG :: "('a => 'b => 'b) => 'b => 'a multiset => 'b => bool"
for f :: "'a => 'b => 'b"
and z :: 'b
where
emptyI [intro]: "fold_msetG f z {#} z"
| insertI [intro]: "fold_msetG f z A y ==> fold_msetG f z (A + {#x#}) (f x y)"


inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y"

definition
fold_mset :: "('a => 'b => 'b) => 'b => 'a multiset => 'b" where
"fold_mset f z A = (THE x. fold_msetG f z A x)"


lemma Diff1_fold_msetG:
"fold_msetG f z (A - {#x#}) y ==> x ∈# A ==> fold_msetG f z A (f x y)"

apply (frule_tac x = x in fold_msetG.insertI)
apply auto
done

lemma fold_msetG_nonempty: "∃x. fold_msetG f z A x"
apply (induct A)
apply blast
apply clarsimp
apply (drule_tac x = x in fold_msetG.insertI)
apply auto
done

lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
unfolding fold_mset_def by blast

context fun_left_comm
begin


lemma fold_msetG_determ:
"fold_msetG f z A x ==> fold_msetG f z A y ==> y = x"

proof (induct arbitrary: x y z rule: full_multiset_induct)
case (less M x1 x2 Z)
have IH: "∀A. A < M -->
(∀x x' x''. fold_msetG f x'' A x --> fold_msetG f x'' A x'
--> x' = x)"
by fact
have Mfoldx1: "fold_msetG f Z M x1" and Mfoldx2: "fold_msetG f Z M x2" by fact+
show ?case
proof (rule fold_msetG.cases [OF Mfoldx1])
assume "M = {#}" and "x1 = Z"
then show ?case using Mfoldx2 by auto
next
fix B b u
assume "M = B + {#b#}" and "x1 = f b u" and Bu: "fold_msetG f Z B u"
then have MBb: "M = B + {#b#}" and x1: "x1 = f b u" by auto
show ?case
proof (rule fold_msetG.cases [OF Mfoldx2])
assume "M = {#}" "x2 = Z"
then show ?case using Mfoldx1 by auto
next
fix C c v
assume "M = C + {#c#}" and "x2 = f c v" and Cv: "fold_msetG f Z C v"
then have MCc: "M = C + {#c#}" and x2: "x2 = f c v" by auto
then have CsubM: "C < M" by simp
from MBb have BsubM: "B < M" by simp
show ?case
proof cases
assume "b=c"
then moreover have "B = C" using MBb MCc by auto
ultimately show ?thesis using Bu Cv x1 x2 CsubM IH by auto
next
assume diff: "b ≠ c"
let ?D = "B - {#c#}"
have cinB: "c ∈# B" and binC: "b ∈# C" using MBb MCc diff
by (auto intro: insert_noteq_member dest: sym)
have "B - {#c#} < B" using cinB by (rule mset_less_diff_self)
then have DsubM: "?D < M" using BsubM by (blast intro: order_less_trans)
from MBb MCc have "B + {#b#} = C + {#c#}" by blast
then have [simp]: "B + {#b#} - {#c#} = C"
using MBb MCc binC cinB by auto
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
using MBb MCc diff binC cinB
by (auto simp: multiset_add_sub_el_shuffle)
then obtain d where Dfoldd: "fold_msetG f Z ?D d"
using fold_msetG_nonempty by iprover
then have "fold_msetG f Z B (f c d)" using cinB
by (rule Diff1_fold_msetG)
then have "f c d = u" using IH BsubM Bu by blast
moreover
have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
by (auto simp: multiset_add_sub_el_shuffle
dest: fold_msetG.insertI [where x=b])

then have "f b d = v" using IH CsubM Cv by blast
ultimately show ?thesis using x1 x2
by (auto simp: fun_left_comm)
qed
qed
qed
qed

lemma fold_mset_insert_aux:
"(fold_msetG f z (A + {#x#}) v) =
(∃y. fold_msetG f z A y ∧ v = f x y)"

apply (rule iffI)
prefer 2
apply blast
apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
apply (blast intro: fold_msetG_determ)
done

lemma fold_mset_equality: "fold_msetG f z A y ==> fold_mset f z A = y"
unfolding fold_mset_def by (blast intro: fold_msetG_determ)

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

apply (simp add: fold_mset_def fold_mset_insert_aux add_commute)
apply (rule the_equality)
apply (auto cong add: conj_cong
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)

done

lemma fold_mset_insert_idem:
"fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"

apply (simp add: fold_mset_def fold_mset_insert_aux)
apply (rule the_equality)
apply (auto cong add: conj_cong
simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)

done

lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
by (induct A) (auto simp: fold_mset_insert fun_left_comm [of x])

lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
using fold_mset_insert [of z "{#}"] by simp

lemma fold_mset_union [simp]:
"fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"

proof (induct A)
case empty then show ?case by simp
next
case (add A x)
have "A + {#x#} + B = (A+B) + {#x#}" by (simp add: add_ac)
then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))"
by (simp add: fold_mset_insert)
also have "… = fold_mset f (fold_mset f z (A + {#x#})) B"
by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
finally show ?case .
qed

lemma fold_mset_fusion:
assumes "fun_left_comm g"
shows "(!!x y. h (g x y) = f x (h y)) ==> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")

proof -
interpret fun_left_comm g by (fact assms)
show "PROP ?P" by (induct A) auto
qed

lemma fold_mset_rec:
assumes "a ∈# A"
shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"

proof -
from assms obtain A' where "A = A' + {#a#}"
by (blast dest: multi_member_split)
then show ?thesis by simp
qed

end

text {*
A note on code generation: When defining some function containing a
subterm @{term"fold_mset F"}, code generation is not automatic. When
interpreting locale @{text left_commutative} with @{text F}, the
would be code thms for @{const fold_mset} become thms like
@{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
contains defined symbols, i.e.\ is not a code thm. Hence a separate
constant with its own code thms needs to be introduced for @{text
F}. See the image operator below.
*}



subsection {* Image *}

definition image_mset :: "('a => 'b) => 'a multiset => 'b multiset" where
"image_mset f = fold_mset (op + o single o f) {#}"


interpretation image_left_comm: fun_left_comm "op + o single o f"
proof qed (simp add: add_ac)

lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
by (simp add: image_mset_def)

lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
by (simp add: image_mset_def)

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

by (simp add: image_mset_def add_ac)

lemma image_mset_union [simp]:
"image_mset f (M+N) = image_mset f M + image_mset f N"

apply (induct N)
apply simp
apply (simp add: add_assoc [symmetric] image_mset_insert)
done

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

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

syntax
"_comprehension1_mset" :: "'a => 'b => 'b multiset => 'a multiset"
("({#_/. _ :# _#})")

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


syntax
"_comprehension2_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#}"}.
*}



subsection {* 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 [code del]: "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_of A, set_of 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_of A, set_of 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_of A, set_of 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_of A, set_of B) ∈ max_strict ∨ (B = {#} ∧ A = {#})"

by auto
from pw_leq_step have "x = y ∨ (x, y) ∈ pair_less"
unfolding pair_leq_def by auto
thus ?case
proof
assume [simp]: "x = y"
have
"{#x#} + X = A + ({#y#}+Z)
∧ {#y#} + Y = B + ({#y#}+Z)
∧ ((set_of A, set_of B) ∈ max_strict ∨ (B = {#} ∧ A = {#}))"

by (auto simp: add_ac)
thus ?case by (intro exI)
next
assume A: "(x, y) ∈ pair_less"
let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
have "{#x#} + X = ?A' + Z"
"{#y#} + Y = ?B' + Z"

by (auto simp add: add_ac)
moreover have
"(set_of ?A', set_of ?B') ∈ max_strict"

using 1 A unfolding max_strict_def
by (auto elim!: max_ext.cases)
ultimately show ?thesis by blast
qed
qed

lemma
assumes pwleq: "pw_leq Z Z'"
shows ms_strictI: "(set_of A, set_of B) ∈ max_strict ==> (Z + A, Z' + B) ∈ ms_strict"
and ms_weakI1: "(set_of A, set_of 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_of A', set_of B') ∈ max_strict ∨ (A' = {#} ∧ B' = {#})"

by blast
{
assume max: "(set_of A, set_of B) ∈ max_strict"
from mx_or_empty
have "(Z'' + (A + A'), Z'' + (B + B')) ∈ ms_strict"
proof
assume max': "(set_of A', set_of B') ∈ max_strict"
with max have "(set_of (A + A'), set_of (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:add_ac)
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:add_ac)
qed

lemma empty_idemp: "{#} + 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 m i =
(if m <= 0 then
rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
else
rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)

val mset_nonempty_tac =
rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}

val regroup_munion_conv =
Function_Lib.regroup_conv @{const_abbrev Mempty} @{const_name plus}
(map (fn t => t RS eq_reflection) (@{thms add_ac} @ @{thms empty_idemp}))

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

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



subsection {* Legacy theorem bindings *}

lemmas multi_count_eq = multiset_ext_iff [symmetric]

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

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

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

lemmas union_ac = union_assoc union_commute union_lcomm

lemma union_right_cancel: "M + K = N + K <-> M = (N::'a multiset)"
by (fact add_right_cancel)

lemma union_left_cancel: "K + M = K + N <-> M = (N::'a multiset)"
by (fact add_left_cancel)

lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y ==> X = Y"
by (fact add_imp_eq)

lemma mset_less_trans: "(M::'a multiset) < K ==> K < N ==> M < N"
by (fact order_less_trans)

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

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

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

lemmas multiset_inter_ac =
multiset_inter_commute
multiset_inter_assoc
multiset_inter_left_commute


lemma mult_less_not_refl:
"¬ 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)

ML {*
fun multiset_postproc _ maybe_name all_values (T as Type (_, [elem_T]))
(Const _ $ t') =
let
val (maybe_opt, ps) =
Nitpick_Model.dest_plain_fun t' ||> op ~~
||> map (apsnd (snd o HOLogic.dest_number))
fun elems_for t =
case AList.lookup (op =) ps t of
SOME n => replicate n t
| NONE => [Const (maybe_name, elem_T --> elem_T) $ t]
in
case maps elems_for (all_values elem_T) @
(if maybe_opt then [Const (Nitpick_Model.unrep (), elem_T)]
else []) of
[] => Const (@{const_name zero_class.zero}, T)
| ts => foldl1 (fn (t1, t2) =>
Const (@{const_name plus_class.plus}, T --> T --> T)
$ t1 $ t2)
(map (curry (op $) (Const (@{const_name single},
elem_T --> T))) ts)
end
| multiset_postproc _ _ _ _ t = t
*}


setup {*
Nitpick.register_term_postprocessor @{typ "'a multiset"} multiset_postproc
*}


end