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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra resring.spad}
+\author{Patrizia Gianni}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{domain RESRING ResidueRing}
+<<domain RESRING ResidueRing>>=
+)abbrev domain RESRING ResidueRing
+++ Author: P.Gianni
+++ Date Created: December 1992
+++ Date Last Updated:
+++ Basic Functions: 
+++ Related Constructors:
+++ Also See:
+++ AMS Classifications:
+++ Keywords: 
+++ References:
+++ Description: ResidueRing is the quotient of a polynomial ring by  an ideal.
+++ The ideal is given as a list of generators. The elements of the domain
+++ are equivalence classes expressed in terms of reduced elements
+
+ResidueRing(F,Expon,VarSet,FPol,LFPol) : Dom  == Body
+ where
+   F       :  Field
+   Expon   :  OrderedAbelianMonoidSup
+   VarSet  :  OrderedSet
+   FPol    :  PolynomialCategory(F, Expon, VarSet)
+   LFPol   :  List FPol
+   
+   Dom   == Join(CommutativeRing, Algebra F) with
+     reduce   :   FPol -> $
+     ++ reduce(f) produces the equivalence class of f in the residue ring
+     coerce   :   FPol -> $
+     ++ coerce(f) produces the equivalence class of f in the residue ring
+     lift     :     $  -> FPol
+     ++ lift(x) return the canonical representative of the equivalence class x
+   Body  ==  add 
+    --representation
+      Rep:= FPol
+      import GroebnerPackage(F,Expon,VarSet,FPol)
+      relations:= groebner(LFPol)
+      relations = [1] => error "the residue ring is the zero ring"
+    --declarations
+      x,y: $
+    --definitions
+      0 == 0$Rep
+      1 == 1$Rep
+      reduce(f : FPol) : $ == normalForm(f,relations)
+      coerce(f : FPol) : $ == normalForm(f,relations)
+      lift x  == x :: Rep :: FPol
+      x + y == x +$Rep y
+      -x == -$Rep x
+      x*y == normalForm(lift(x *$Rep y),relations)
+      (n : Integer) * x == n *$Rep x
+      (a : F) * x == a *$Rep x
+      x = y == x =$Rep y
+      characteristic()      == characteristic()$F
+      coerce(x) : OutputForm == coerce(x)$Rep
+
+@
+\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 RESRING ResidueRing>>
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}