\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{src/algebra retract.spad}
\author{Manuel Bronstein}

\maketitle

\begin{abstract}
\end{abstract}
\tableofcontents
\eject

\section{category FRETRCT FullyRetractableTo}

<<category FRETRCT FullyRetractableTo>>=
import Type
import RetractableTo
import Integer
import Fraction
)abbrev category FRETRCT FullyRetractableTo
++ Author: Manuel Bronstein
++ Description:
++   A is fully retractable to B means that A is retractable to B, and,
++   in addition, if B is retractable to the integers or rational
++   numbers then so is A.
++   In particular, what we are asserting is that there are no integers
++   (rationals) in A which don't retract into B.
++ Date Created: March 1990
++ Date Last Updated: 9 April 1991
FullyRetractableTo(S: Type): Category == RetractableTo(S) with
    if (S has RetractableTo Integer) then RetractableTo Integer
    if (S has RetractableTo Fraction Integer) then
              RetractableTo Fraction Integer
  add
    if not(S is Integer) then
      if (S has RetractableTo Integer) then    -- induction
        coerce(n:Integer):%  == n::S::%
        retract(r:%):Integer == retract(retract(r)@S)
 
        retractIfCan(r:%):Union(Integer, "failed") ==
          (u:= retractIfCan(r)@Union(S,"failed")) case "failed"=> "failed"
          retractIfCan(u::S)
 
    if not(S is Fraction Integer) then
      if (S has RetractableTo Fraction Integer) then   -- induction
        coerce(n:Fraction Integer):%  == n::S::%
        retract(r:%):Fraction(Integer) == retract(retract(r)@S)
 
        retractIfCan(r:%):Union(Fraction Integer, "failed") ==
          (u:=retractIfCan(r)@Union(S,"failed")) case "failed"=>"failed"
          retractIfCan(u::S)

@

\section{package INTRET IntegerRetractions}

<<package INTRET IntegerRetractions>>=
import RetractableTo 
import Boolean
import Integer
)abbrev package INTRET IntegerRetractions
++ Author: Manuel Bronstein
++ Description: Provides integer testing and retraction functions.
++ Date Created: March 1990
++ Date Last Updated: 9 April 1991
IntegerRetractions(S:RetractableTo(Integer)): with
    integer     : S -> Integer
      ++ integer(x) returns x as an integer;
      ++ error if x is not an integer;
    integer?    : S -> Boolean
      ++ integer?(x) is true if x is an integer, false otherwise;
    integerIfCan: S -> Union(Integer, "failed")
      ++ integerIfCan(x) returns x as an integer,
      ++ "failed" if x is not an integer;
  == add
    integer s      == retract s
    integer? s     == retractIfCan(s) case Integer
    integerIfCan s == retractIfCan s

@

\section{package RATRET RationalRetractions}

<<package RATRET RationalRetractions>>=
import RetractableTo
import Boolean
import Integer
import Fraction
)abbrev package RATRET RationalRetractions
++ Author: Manuel Bronstein
++ Description: rational number testing and retraction functions.
++ Date Created: March 1990
++ Date Last Updated: 9 April 1991
RationalRetractions(S:RetractableTo(Fraction Integer)): with
    rational     : S -> Fraction Integer
      ++ rational(x) returns x as a rational number;
      ++ error if x is not a rational number;
    rational?    : S -> Boolean
      ++ rational?(x) returns true if x is a rational number,
      ++ false otherwise;
    rationalIfCan: S -> Union(Fraction Integer, "failed")
      ++ rationalIfCan(x) returns x as a rational number,
      ++ "failed" if x is not a rational number;
  == add
    rational s      == retract s
    rational? s     == retractIfCan(s) case Fraction(Integer)
    rationalIfCan s == retractIfCan s

@

\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 FRETRCT FullyRetractableTo>>
<<package INTRET IntegerRetractions>>
<<package RATRET RationalRetractions>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
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\end{document}