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product.spad
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product.spad
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)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
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
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-- distribution.
--
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--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
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