)abbrev domain PRODUCT Product
++ Description:
++  This domain implements cartesian product. If the underlying
++  domains are both Finite then the resulting Product is also Finite
++  and can be enumerated via size(), index(), location(), etc.
++  The index of the second component (B) varies most quickly.
Product (A : SetCategory, B : SetCategory) : C == T
 where
  C == SetCategory  with
       if A has Finite and B has Finite then Finite
       if A has Monoid and B has Monoid then Monoid
       if A has AbelianMonoid and B has AbelianMonoid then AbelianMonoid
       if A has CancellationAbelianMonoid and
          B has CancellationAbelianMonoid then CancellationAbelianMonoid
       if A has Group  and B has Group  then  Group
       if A has AbelianGroup and B has AbelianGroup then  AbelianGroup
       if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup
                                             then OrderedAbelianMonoidSup
       if A has OrderedSet and B has OrderedSet then  OrderedSet
       if A has Comparable and B has Comparable then  Comparable
       if A has Hashable and B has Hashable then Hashable

       construct     : (A, B) -> %
        ++ construct(a, b) creates element of the product with
        ++ components a and b.
       first  :   %   -> A
        ++ first(x) selects first component of the product
       second :   %   -> B
        ++ second(x) selects second component of the product

  T == add

    --representations
       Rep := Record(acomp : A, bcomp : B)

    --declarations
       x, y : %
       i : NonNegativeInteger
       p : NonNegativeInteger
       a : A
       b : B
       d : Integer

    --define
       coerce(x) : OutputForm == bracket [(x.acomp)::OutputForm,
                                        (x.bcomp)::OutputForm]
       x = y ==
           x.acomp = y.acomp => x.bcomp = y.bcomp
           false
       construct(a : A, b : B) : %   == [a, b]$Rep

       first(x : %) : A   == x.acomp

       second (x : %) : B == x.bcomp

       if A has Monoid and B has Monoid then
          1 == [1$A, 1$B]
          x * y == [x.acomp * y.acomp, x.bcomp * y.bcomp]
          x ^ p == [x.acomp ^ p , x.bcomp ^ p]

       if A has Finite and B has Finite then
          size() == size()$A * size()$B
          index(n) == [index((((n::Integer-1) quo size()$B )+1)
                         ::PositiveInteger)$A,
                       index((((n::Integer-1) rem size()$B )+1)
                         ::PositiveInteger)$B]
          random() == [random()$A, random()$B]
          lookup(x) == ((lookup(x.acomp)$A::Integer-1) * size()$B::Integer +
                        lookup(x.bcomp)$B::Integer)::PositiveInteger

       if A has Hashable and B has Hashable then

           hashUpdate!(s : HashState, x : %) : HashState ==
               hashUpdate!(hashUpdate!(s, first(x)), second(x))

       if A has Group and B has Group then
          inv(x) == [inv(x.acomp), inv(x.bcomp)]

       if A has AbelianMonoid and B has AbelianMonoid then
          0 == [0$A, 0$B]

          x + y == [x.acomp + y.acomp, x.bcomp + y.bcomp]

          c : NonNegativeInteger * x == [c * x.acomp, c*x.bcomp]

       if A has CancellationAbelianMonoid and
          B has CancellationAbelianMonoid then
            subtractIfCan(x, y) : Union(%,"failed") ==
              (na:= subtractIfCan(x.acomp, y.acomp)) case "failed" => "failed"
              (nb:= subtractIfCan(x.bcomp, y.bcomp)) case "failed" => "failed"
              [na@A, nb@B]

       if A has AbelianGroup and B has AbelianGroup then
          - x == [- x.acomp, -x.bcomp]
          (x - y) : % == [x.acomp - y.acomp, x.bcomp - y.bcomp]
          d * x == [d * x.acomp, d * x.bcomp]

       if A has OrderedAbelianMonoidSup and B has OrderedAbelianMonoidSup then
          sup(x, y) == [sup(x.acomp, y.acomp), sup(x.bcomp, y.bcomp)]

       if A has OrderedSet and B has OrderedSet then
          x < y ==
               xa:= x.acomp ; ya:= y.acomp
               xa < ya => true
               xb:= x.bcomp ; yb:= y.bcomp
               xa = ya => (xb < yb)
               false
       if A has Comparable and B has Comparable then
          smaller?(x, y) ==
               xa := x.acomp ; ya:= y.acomp
               smaller?(xa, ya) => true
               xb := x.bcomp ; yb:= y.bcomp
               xa = ya => smaller?(xb, yb)
               false


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