)abbrev domain MSET Multiset
++ Author: Stephen M. Watt, William H. Burge, Richard D. Jenks, Frederic Lehobey
++ Date Created: NK
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: A multiset is a set with multiplicities.
Multiset(S : SetCategory) : Join(MultisetAggregate S, finiteAggregate, _
          shallowlyMutable) with
        multiset : () -> %
          ++ multiset()$D creates an empty multiset of domain D.
        multiset : S -> %
          ++ multiset(s) creates a multiset with singleton s.
        multiset : List S -> %
          ++ multiset(ls) creates a multiset with elements from \spad{ls}.
        remove : (S, %, Integer) -> %
          ++ remove(x, ms, number) removes at most \spad{number} copies of
          ++ element x if \spad{number} is positive, all of them if
          ++ \spad{number} equals zero, and all but at most \spad{-number} if
          ++ \spad{number} is negative.
        remove : (S -> Boolean, %, Integer) -> %
          ++ remove(p, ms, number) removes at most \spad{number} copies of
          ++ elements x such that \spad{p(x)} is \spadfun{true}
          ++ if \spad{number} is positive, all of them if
          ++ \spad{number} equals zero, and all but at most \spad{-number} if
          ++ \spad{number} is negative.
        remove! : (S, %, Integer) -> %
          ++ remove!(x, ms, number) removes destructively at most \spad{number}
          ++ copies of element x if \spad{number} is positive, all
          ++ of them if \spad{number} equals zero, and all but at most
          ++ \spad{-number} if \spad{number} is negative.
        remove! : (S -> Boolean, %, Integer) -> %
          ++ remove!(p, ms, number) removes destructively at most \spad{number}
          ++ copies of elements x such that \spad{p(x)} is
          ++ \spadfun{true} if \spad{number} is positive, all of them if
          ++ \spad{number} equals zero, and all but at most \spad{-number} if
          ++ \spad{number} is negative.

    == add

        Tbl ==> Table(S, Integer)
        tbl ==> table$Tbl
        Rep := Record(count : Integer, table : Tbl)

        n : Integer
        ms, m1, m2 : %
        t,  t1, t2 : Tbl
        D ==> Record(entry : S, count : NonNegativeInteger)
        K ==> Record(key : S, entry : Integer)

        elt(t : Tbl, s : S) : Integer ==
          a := search(s, t)$Tbl
          a case "failed" => 0
          a::Integer

        empty() : % == [0, tbl()]
        multiset() : % == empty()
        dictionary() : % == empty()                       -- DictionaryOperations
        set() : % == empty()

        construct(l : List S) : % ==
            t := tbl()
            n := 0
            for e in l repeat
              t.e := inc t.e
              n := inc n
            [n, t]
        multiset(l : List S) : % == construct l
        dictionary(l : List S) : % == construct l           -- DictionaryOperations
        set(l : List S) : % == construct l

        multiset(s : S) : % == construct [s]

        if S has ConvertibleTo InputForm then
          convert(ms : %) : InputForm ==
            convert [convert('multiset)@InputForm,
             convert(parts ms)@InputForm]

        members(ms : %) : List S == keys ms.table

        coerce(ms : %) : OutputForm ==
            l : List OutputForm := empty()
            t := ms.table
            colon := message(" : ")
            for e in keys t repeat
                ex := e::OutputForm
                n := t.e
                item :=
                  n > 1 => hconcat [n :: OutputForm, colon, ex]
                  ex
                l := cons(item, l)
            brace l

        duplicates(ms : %) : List D ==                      -- MultiDictionary
          ld : List D := empty()
          t := ms.table
          for e in keys t | (n := t.e) > 1 repeat
            ld := cons([e, n::NonNegativeInteger], ld)
          ld

        extract!(ms : %) : S ==                            -- BagAggregate
          empty? ms => error "extract: Empty multiset"
          ms.count := dec ms.count
          t := ms.table
          e := inspect(t).key
          if (n := t.e) > 1 then t.e := dec n
           else remove!(e, t)
          e

        inspect(ms : %) : S == inspect(ms.table).key        -- BagAggregate

        insert!(e : S, ms : %) : % ==                         -- BagAggregate
            ms.count   := inc ms.count
            ms.table.e := inc ms.table.e
            ms

        member?(e : S, ms : %) : Boolean == member?(e, keys ms.table)

        empty?(ms : %) : Boolean == ms.count = 0

        #(ms : %) : NonNegativeInteger == ms.count::NonNegativeInteger

        count(e : S, ms : %) : NonNegativeInteger == ms.table.e::NonNegativeInteger

        remove!(e : S, ms : %, max : Integer) : % ==
          zero? max => remove!(e, ms)
          t := ms.table
          if member?(e, keys t) then
            ((n := t.e) <= max) =>
              remove!(e, t)
              ms.count := ms.count-n
            max > 0 =>
              t.e := n-max
              ms.count := ms.count-max
            (n := n+max) > 0 =>
              t.e := -max
              ms.count := ms.count-n
          ms

        remove!(p : S -> Boolean, ms : %, max : Integer) : % ==
          zero? max => remove!(p, ms)
          t := ms.table
          for e in keys t | p(e) repeat
            ((n := t.e) <= max) =>
              remove!(e, t)
              ms.count := ms.count-n
            max > 0 =>
              t.e := n-max
              ms.count := ms.count-max
            (n := n+max) > 0 =>
              t.e := -max
              ms.count := ms.count-n
          ms

        remove(e : S, ms : %, max : Integer) : % == remove!(e, copy ms, max)

        remove(p : S -> Boolean, ms : %, max : Integer) : % == remove!(p, copy ms, max)

        remove!(e : S, ms : %) : % ==                        -- DictionaryOperations
          t := ms.table
          if member?(e, keys t) then
            ms.count := ms.count-t.e
            remove!(e, t)
          ms

        remove!(p : S ->Boolean, ms : %) : % ==              -- DictionaryOperations
          t := ms.table
          for e in keys t | p(e) repeat
            ms.count := ms.count-t.e
            remove!(e, t)
          ms

        select!(p : S -> Boolean, ms : %) : % ==            -- DictionaryOperations
          remove!((s1 : S) : Boolean +-> not p(s1), ms)

        removeDuplicates!(ms : %) : % ==                   -- MultiDictionary
          t := ms.table
          l := keys t
          for e in l repeat t.e := 1
          ms.count := #l
          ms

        insert!(e : S, ms : %, more : NonNegativeInteger) : % == -- MultiDictionary
            ms.count   := ms.count+more
            ms.table.e := ms.table.e+more
            ms

        map!(f : S->S, ms : %) : % ==                       -- HomogeneousAggregate
          t := ms.table
          t1 := tbl()
          for e in keys t repeat
            t1.f(e) := t.e
            remove!(e, t)
          ms.table := t1
          ms

        map(f : S -> S, ms : %) : % == map!(f, copy ms)     -- HomogeneousAggregate

        parts(m : %) : List S ==
          l := empty()$List(S)
          t := m.table
          for e in keys t repeat
            for i in 1..t.e repeat
              l := cons(e, l)
          l

        union(m1 : %, m2 : %) : % ==
            t := tbl()
            t1:= m1.table
            t2:= m2.table
            for e in keys t1 repeat t.e := t1.e
            for e in keys t2 repeat t.e := t2.e + t.e
            [m1.count + m2.count, t]

        intersect(m1 : %, m2 : %) : % ==
--          if #m1 > #m2 then intersect(m2, m1)
            t := tbl()
            t1:= m1.table
            t2:= m2.table
            n := 0
            for e in keys t1 repeat
              m := min(t1.e, t2.e)
              m > 0 =>
                m := t1.e + t2.e
                t.e := m
                n := n + m
            [n, t]

        difference(m1 : %, m2 : %) : % ==
            t := tbl()
            t1:= m1.table
            t2:= m2.table
            n := 0
            for e in keys t1 repeat
              k1 := t1.e
              k2 := t2.e
              k1 > 0 and k2 = 0 =>
                t.e := k1
                n := n + k1
            n = 0 => empty()
            [n, t]

        symmetricDifference(m1 : %, m2 : %) : % ==
            union(difference(m1, m2), difference(m2, m1))

        m1 = m2 ==
            m1.count ~= m2.count => false
            t1 := m1.table
            t2 := m2.table
            for e in keys t1 repeat
                t1.e ~= t2.e => return false
            for e in keys t2 repeat
                t1.e ~= t2.e => return false
            true

        m1 < m2 ==
            m1.count >= m2.count => false
            t1 := m1.table
            t2 := m2.table
            for e in keys t1 repeat
                t1.e > t2.e => return false
            m1.count < m2.count

        subset?(m1 : %, m2 : %) : Boolean ==
            m1.count > m2.count => false
            t1 := m1.table
            t2 := m2.table
            for e in keys t1 repeat t1.e > t2.e => return false
            true

        m1 <= m2 == subset?(m1, m2)

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