path: root/src/algebra/indexedp.spad.pamphlet

\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra indexedp.spad}
\author{James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category IDPC IndexedDirectProductCategory}
<<category IDPC IndexedDirectProductCategory>>=
)abbrev category IDPC IndexedDirectProductCategory
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This category represents the direct product of some set with
++ respect to an ordered indexing set.

IndexedDirectProductCategory(A:SetCategory,S:OrderedSet): Category ==
  SetCategory with
    map:           (A -> A, %) -> %
       ++ map(f,z) returns the new element created by applying the
       ++ function f to each component of the direct product element z.
    monomial:         (A, S) -> %
       ++ monomial(a,s) constructs a direct product element with the s
       ++ component set to \spad{a}
    leadingCoefficient:   % -> A
       ++ leadingCoefficient(z) returns the coefficient of the leading
       ++ (with respect to the ordering on the indexing set)
       ++ monomial of z.
       ++ Error: if z has no support.
    leadingSupport:   % -> S
       ++ leadingSupport(z) returns the index of leading
       ++ (with respect to the ordering on the indexing set) monomial of z.
       ++ Error: if z has no support.
    reductum:      % -> %
       ++ reductum(z) returns a new element created by removing the
       ++ leading coefficient/support pair from the element z.
       ++ Error: if z has no support.

@
\section{domain IDPO IndexedDirectProductObject}
<<domain IDPO IndexedDirectProductObject>>=
)abbrev domain IDPO IndexedDirectProductObject
++ Indexed direct products of objects over a set \spad{A}
++ of generators indexed by an ordered set S. All items have finite support.
IndexedDirectProductObject(A:SetCategory,S:OrderedSet): IndexedDirectProductCategory(A,S)
 == add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
    --declarations
       x,y: %
       f: A -> A
       s: S
    --define
       x = y ==
         while not null x and not null y repeat
           x.first.k ~= y.first.k => return false
           x.first.c ~= y.first.c => return false
           x:=x.rest
           y:=y.rest
         null x and null y

       coerce(x:%):OutputForm ==
          bracket [rarrow(t.k :: OutputForm, t.c :: OutputForm) for t in x]

       -- sample():% == [[sample()$S,sample()$A]$Term]$Rep

       monomial(r,s) == [[s,r]]
       map(f,x) == [[tm.k,f(tm.c)] for tm in x]

       reductum x     ==
          rest x
       leadingCoefficient x  ==
          null x => error "Can't take leadingCoefficient of empty product element"
          x.first.c
       leadingSupport x  ==
          null x => error "Can't take leadingCoefficient of empty product element"
          x.first.k

@
\section{domain IDPAM IndexedDirectProductAbelianMonoid}
<<domain IDPAM IndexedDirectProductAbelianMonoid>>=
)abbrev domain IDPAM IndexedDirectProductAbelianMonoid
++ Indexed direct products of abelian monoids over an abelian monoid \spad{A} of
++ generators indexed by the ordered set S. All items have finite support.
++ Only non-zero terms are stored.
IndexedDirectProductAbelianMonoid(A:AbelianMonoid,S:OrderedSet):
    Join(AbelianMonoid,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductObject(A,S) add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
       x,y: %
       r: A
       n: NonNegativeInteger
       f: A -> A
       s: S
       0  == []
       zero? x ==  null x

	-- PERFORMANCE CRITICAL; Should build list up
	--  by merging 2 sorted lists.   Doing this will
	-- avoid the recursive calls (very useful if there is a
	-- large number of vars in a polynomial.
--       x + y  ==
--          null x => y
--          null y => x
--          y.first.k > x.first.k => cons(y.first,(x + y.rest))
--          x.first.k > y.first.k => cons(x.first,(x.rest + y))
--          r:= x.first.c + y.first.c
--          r = 0 => x.rest + y.rest
--          cons([x.first.k,r],(x.rest + y.rest))
       qsetrest!: (Rep, Rep) -> Rep
       qsetrest!(l: Rep, e: Rep): Rep == RPLACD(l, e)$Lisp

       x + y == 
	 	null x => y
		null y => x
		endcell: Rep := empty()
		res:  Rep := empty()
		while not empty? x and not empty? y repeat 
			newcell := empty()
			if x.first.k = y.first.k then
				r:= x.first.c + y.first.c
				if not zero? r then 
					newcell := cons([x.first.k, r], empty())
				x := rest x
				y := rest y
			else if x.first.k > y.first.k then
				newcell := cons(x.first, empty())
				x := rest x
			else
				newcell := cons(y.first, empty())
				y := rest y
			if not empty? newcell then 
				if not empty? endcell then
					qsetrest!(endcell, newcell)
					endcell := newcell
				else
					res     := newcell;
					endcell := res
		if empty? x then end := y
		else end := x
		if empty? res then res := end
		else qsetrest!(endcell, end)
		res

       n * x  ==
             n = 0 => 0
             n = 1 => x
             [[u.k,a] for u in x | (a:=n*u.c) ~= 0$A]

       monomial(r,s) == (r = 0 => 0; [[s,r]])
       map(f,x) == [[tm.k,a] for tm in x | (a:=f(tm.c)) ~= 0$A]

       reductum x     == (null x => 0; rest x)
       leadingCoefficient x  == (null x => 0; x.first.c)

@
\section{domain IDPOAM IndexedDirectProductOrderedAbelianMonoid}
<<domain IDPOAM IndexedDirectProductOrderedAbelianMonoid>>=
)abbrev domain IDPOAM IndexedDirectProductOrderedAbelianMonoid
++ Indexed direct products of ordered abelian monoids \spad{A} of
++ generators indexed by the ordered set S.
++ The inherited order is lexicographical.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoid(A:OrderedAbelianMonoid,S:OrderedSet):
    Join(OrderedAbelianMonoid,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductAbelianMonoid(A,S) add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
       x,y: %
       x<y ==
         empty? y => false
         empty? x => true   -- note careful order of these two lines
         y.first.k > x.first.k => true
         y.first.k < x.first.k => false
         y.first.c > x.first.c => true
         y.first.c < x.first.c => false
         x.rest < y.rest

@
\section{domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup}
<<domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup>>=
)abbrev domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup
++ Indexed direct products of ordered abelian monoid sups \spad{A},
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductOrderedAbelianMonoidSup(A:OrderedAbelianMonoidSup,S:OrderedSet):
    Join(OrderedAbelianMonoidSup,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductOrderedAbelianMonoid(A,S) add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
       x,y: %
       r: A
       s: S

       subtractIfCan(x,y) ==
         empty? y => x
         empty? x => "failed"
         x.first.k < y.first.k => "failed"
         x.first.k > y.first.k =>
             t:= subtractIfCan(x.rest, y)
             t case "failed" => "failed"
             cons( x.first, t)
         u:=subtractIfCan(x.first.c, y.first.c)
         u case "failed" => "failed"
         zero? u => subtractIfCan(x.rest, y.rest)
         t:= subtractIfCan(x.rest, y.rest)
         t case "failed" => "failed"
         cons([x.first.k,u],t)

       sup(x,y) ==
         empty? y => x
         empty? x => y
         x.first.k < y.first.k => cons(y.first,sup(x,y.rest))
         x.first.k > y.first.k => cons(x.first,sup(x.rest,y))
         u:=sup(x.first.c, y.first.c)
         cons([x.first.k,u],sup(x.rest,y.rest))

@
\section{domain IDPAG IndexedDirectProductAbelianGroup}
<<domain IDPAG IndexedDirectProductAbelianGroup>>=
)abbrev domain IDPAG IndexedDirectProductAbelianGroup
++ Indexed direct products of abelian groups over an abelian group \spad{A} of
++ generators indexed by the ordered set S.
++ All items have finite support: only non-zero terms are stored.
IndexedDirectProductAbelianGroup(A:AbelianGroup,S:OrderedSet):
    Join(AbelianGroup,IndexedDirectProductCategory(A,S))
 ==  IndexedDirectProductAbelianMonoid(A,S) add
    --representations
       Term:=  Record(k:S,c:A)
       Rep:=  List Term
       x,y: %
       r: A
       n: Integer
       f: A -> A
       s: S
       -x == [[u.k,-u.c] for u in x]
       n * x  ==
             n = 0 => 0
             n = 1 => x
             [[u.k,a] for u in x | (a:=n*u.c) ~= 0$A]

       qsetrest!: (Rep, Rep) -> Rep
       qsetrest!(l: Rep, e: Rep): Rep == RPLACD(l, e)$Lisp

       x - y ==
                null x => -y
                null y => x
                endcell: Rep := empty()
                res:  Rep := empty()
                while not empty? x and not empty? y repeat
                        newcell := empty()
                        if x.first.k = y.first.k then
                                r:= x.first.c - y.first.c
                                if not zero? r then
                                        newcell := cons([x.first.k, r], empty())
                                x := rest x
                                y := rest y
                        else if x.first.k > y.first.k then
                                newcell := cons(x.first, empty())
                                x := rest x
                        else
                                newcell := cons([y.first.k,-y.first.c], empty())
                                y := rest y
                        if not empty? newcell then
                                if not empty? endcell then
                                        qsetrest!(endcell, newcell)
                                        endcell := newcell
                                else
                                        res     := newcell;
                                        endcell := res
                if empty? x then end := - y
                else end := x
                if empty? res then res := end
                else qsetrest!(endcell, end)
                res

--       x - y  ==
--          empty? x => - y
--          empty? y => x
--          y.first.k > x.first.k => cons([y.first.k,-y.first.c],(x - y.rest))
--          x.first.k > y.first.k => cons(x.first,(x.rest - y))
--          r:= x.first.c - y.first.c
--          r = 0 => x.rest - y.rest
--          cons([x.first.k,r],(x.rest - y.rest))

@
\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>>

<<category IDPC IndexedDirectProductCategory>>
<<domain IDPO IndexedDirectProductObject>>
<<domain IDPAM IndexedDirectProductAbelianMonoid>>
<<domain IDPOAM IndexedDirectProductOrderedAbelianMonoid>>
<<domain IDPOAMS IndexedDirectProductOrderedAbelianMonoidSup>>
<<domain IDPAG IndexedDirectProductAbelianGroup>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}