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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra fmod.spad}
\author{The Axiom Team}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain ZMOD IntegerMod}
<<domain ZMOD IntegerMod>>=
)abbrev domain ZMOD IntegerMod
++ Author:
++ Date Created:
++ Date Last Updated: May 29, 2009
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ IntegerMod(n) creates the ring of integers reduced modulo the integer
++ n.
IntegerMod(p:PositiveInteger):
Join(CommutativeRing, Finite, ConvertibleTo Integer, StepThrough) == add
Rep == Integer
size() == p
characteristic == p
lookup x ==
zero? x => p
rep(x)::PositiveInteger
convert(x: %): Integer == rep x
coerce(n: Integer): % == per positiveRemainder(n,p)
coerce(x): OutputForm == rep(x)::OutputForm
0 == per(0$Rep)
1 == per(1$Rep)
init == 0
nextItem(n) ==
m := n+1
m = 0 => "failed"
m
zero? x == rep x = 0
one? x == rep x = 1
x = y ==
rep x = rep y
x:% * y:% ==
per mulmod(rep x,rep y,p)
n:Integer * x:% ==
per mulmod(positiveRemainder(n, p), rep x, p)
x + y ==
per addmod(rep x,rep y,p)
x - y ==
per submod(rep x,rep y,p)
random() == per random(p)$Rep
index a ==
per positiveRemainder(a,p)
-x ==
zero? rep x => 0
per(p - rep x)
x:% ** n:NonNegativeInteger ==
per powmod(rep x, n, p)
recip x ==
(c1, c2, g) := extendedEuclidean(rep x, p)$Rep
not one? g => "failed"
per positiveRemainder(c1, p)
before?(x,y) == before?(rep x,rep y)
@
\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 ZMOD IntegerMod>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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