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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/algebra retract.spad}
+\author{Manuel Bronstein}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{category FRETRCT FullyRetractableTo}
+<<category FRETRCT FullyRetractableTo>>=
+)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>>=
+)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>>=
+)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
+\end{thebibliography}
+\end{document}