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# Edit detail for product.spad revision 1 of 1

 1 Editor: Time: 2007/11/29 04:13:21 GMT-8 Note: missing operations from Finite

changed:
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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra product.spad} \author{The Axiom Team} \maketitle \begin{abstract} This domain implements cartesian product for a pair of (possibly different) domains. 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. \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 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)$B::Integer)::PositiveInteger
hash(x) == hash(x.acomp)$A * size$B::SingleInteger + hash(x.bcomp)$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

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