Changes http://wiki.axiom-developer.org/DirectProduct/diff
--
(from vector.spad.pamphlet)
++added:
+
+\begin{spad}
+)abbrev domain DIRPROD DirectProduct
+++ Author:
+++ Date Created:
+++ Date Last Updated:
+++ Basic Functions:
+++ Related Constructors: Vector, IndexedVector
+++ Also See: OrderedDirectProduct
+++ AMS Classifications:
+++ Keywords:
+++ References:
+++ Description:
+++ This type represents the finite direct or cartesian product of an
+++ underlying component type. This contrasts with simple vectors in that
+++ the members can be viewed as having constant length. Thus many
+++ categorical properties can by lifted from the underlying component type.
+++ Component extraction operations are provided but no updating operations.
+++ Thus new direct product elements can either be created by converting
+++ vector elements using the \spadfun{directProduct} function
+++ or by taking appropriate linear combinations of basis vectors provided
+++ by the \spad{unitVector} operation.
+
+DirectProduct(dim:NonNegativeInteger, R:Type):
+ DirectProductCategory(dim, R) == Vector R add
+
+ Rep := Vector R
+
+ coerce(z:%):Vector(R) == copy(z)$Rep pretend Vector(R)
+ coerce(r:R):% == new(dim, r)$Rep
+
+ parts x == VEC2LIST(x)$Lisp
+
+ directProduct z ==
+ size?(z, dim) => copy(z)$Rep
+ error "Not of the correct length"
+
+
+ if R has SetCategory then
+ same?: % -> Boolean
+ same? z == every?(#1 = z(minIndex z), z)
+
+ x = y == _and/[qelt(x,i)$Rep = qelt(y,i)$Rep for i in 1..dim]
+
+ retract(z:%):R ==
+ same? z => z(minIndex z)
+ error "Not retractable"
+
+ retractIfCan(z:%):Union(R, "failed") ==
+ same? z => z(minIndex z)
+ "failed"
+
+
+ if R has AbelianSemiGroup then
+ u:% + v:% == map(_+ , u, v)$Rep
+
+ if R has AbelianMonoid then
+ 0 == zero(dim)$Vector(R) pretend %
+
+ if R has Monoid then
+ 1 == new(dim, 1)$Vector(R) pretend %
+ u:% * r:R == map(#1 * r, u)
+ r:R * u:% == map(r * #1, u)
+
+
+ if R has CancellationAbelianMonoid then
+ subtractIfCan(u:%, v:%):Union(%,"failed") ==
+ w := new(dim,0)$Vector(R)
+ for i in 1..dim repeat
+ (c := subtractIfCan(qelt(u, i)$Rep, qelt(v,i)$Rep)) case "failed"
=>
+ return "failed"
+ qsetelt_!(w, i, c::R)$Rep
+ w pretend %
+
+ if R has Ring then
+
+ u:% * v:% == map(_* , u, v)$Rep
+
+ recip z ==
+ w := new(dim,0)$Vector(R)
+ for i in minIndex w .. maxIndex w repeat
+ (u := recip qelt(z, i)) case "failed" => return "failed"
+ qsetelt_!(w, i, u::R)
+ w pretend %
+
+ unitVector i ==
+ v:= new(dim,0)$Vector(R)
+ v.i := 1
+ v pretend %
+
+ if R has OrderedSet then
+ x < y ==
+ for i in 1..dim repeat
+ qelt(x,i) < qelt(y,i) => return true
+ qelt(x,i) > qelt(y,i) => return false
+ false
+
+ if R has OrderedAbelianMonoidSup then sup(x, y) == map(sup, x, y)
+
+--)bo $noSubsumption := false
+
+\end{spad}
--
forwarded from http://wiki.axiom-developer.org/[EMAIL PROTECTED]
