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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra product.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain PRODUCT Product}
<<domain PRODUCT Product>>=
)abbrev domain PRODUCT Product
++ Description:
++ This domain implements cartesian product
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
makeprod : (A,B) -> %
++ makeprod(a,b) \undocumented
selectfirst : % -> A
++ selectfirst(x) \undocumented
selectsecond : % -> B
++ selectsecond(x) \undocumented
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 == paren [(x.acomp)::OutputForm,
(x.bcomp)::OutputForm]
x=y ==
x.acomp = y.acomp => x.bcomp = y.bcomp
false
makeprod(a:A,b:B) :% == [a,b]
selectfirst(x:%) : A == x.acomp
selectsecond (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 ()
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
-- coerce(x:%):Symbol ==
-- PrintableForm()
-- formList([x.acomp::Expression,x.bcomp::Expression])$PrintableForm
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<domain PRODUCT Product>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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