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

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra fraction.spad}
\author{Dave Barton, Barry Trager, James Davenport}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain LO Localize}
<<domain LO Localize>>=
)abbrev domain LO Localize
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions: + - / numer denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: localization
++ References:
++ Description: Localize(M,R,S) produces fractions with numerators
++ from an R module M and denominators from some multiplicative subset
++ D of R.
Localize(M:Module R,
         R:CommutativeRing,
         S:SubsetCategory(Monoid, R)): Module R with
      if M has OrderedAbelianGroup then OrderedAbelianGroup
      / :(%,S) -> %
         ++ x / d divides the element x by d.
      / :(M,S) -> %
         ++ m / d divides the element m by d.
      numer: % -> M
         ++ numer x returns the numerator of x.
      denom: % -> S
         ++ denom x returns the denominator of x.
 ==
  add
    --representation
      Rep:= Record(num:M,den:S)
    --declarations
      x,y: %
      n: Integer
      m: M
      r: R
      d: S
    --definitions
      0 == [0,1]
      zero? x == zero? (x.num)
      -x== [-x.num,x.den]
      x=y == y.den*x.num = x.den*y.num
      before?(x,y) == before?(y.den*x.num, x.den*y.num)
      numer x == x.num
      denom x == x.den
      if M has OrderedAbelianGroup then
        x < y == 
--             if y.den::R < 0 then (x,y):=(y,x)
--             if x.den::R < 0 then (x,y):=(y,x)
             y.den*x.num < x.den*y.num
      x+y == [y.den*x.num+x.den*y.num, x.den*y.den]
      n*x == [n*x.num,x.den]
      r*x == if r=x.den then [x.num,1] else [r*x.num,x.den]
      x/d ==
        zero?(u:S:=d*x.den) => error "division by zero"
        [x.num,u]
      m/d == if zero? d then error "division by zero" else [m,d]
      coerce(x:%):OutputForm ==
        one?(xd:=x.den) => (x.num)::OutputForm
        (x.num)::OutputForm / (xd::OutputForm)
      latex(x:%): String ==
        one?(xd:=x.den) => latex(x.num)
        nl : String := concat("{", concat(latex(x.num), "}")$String)$String
        dl : String := concat("{", concat(latex(x.den), "}")$String)$String
        concat("{ ", concat(nl, concat(" \over ", concat(dl, " }")$String)$String)$String)$String

@
\section{domain LA LocalAlgebra}
<<domain LA LocalAlgebra>>=
)abbrev domain LA LocalAlgebra
++ Author: Dave Barton, Barry Trager
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: LocalAlgebra produces the localization of an algebra, i.e.
++ fractions whose numerators come from some R algebra.
LocalAlgebra(A: Algebra R,
             R: CommutativeRing,
             S: SubsetCategory(Monoid, R)): Algebra R with
          if A has OrderedRing then OrderedRing
          / : (%,S) -> %
            ++ x / d divides the element x by d.
          / : (A,S) -> %
            ++ a / d divides the element \spad{a} by d.
          numer: % -> A
            ++ numer x returns the numerator of x.
          denom: % -> S
            ++ denom x returns the denominator of x.
 == Localize(A, R, S) add
        1 == 1$A / 1$S
        x:% * y:% == (numer(x) * numer(y)) / (denom(x) * denom(y))
        characteristic == characteristic$A

@
\section{category QFCAT QuotientFieldCategory}
<<category QFCAT QuotientFieldCategory>>=
)abbrev category QFCAT QuotientFieldCategory
++ Author:
++ Date Created:
++ Date Last Updated: 5th March 1996 
++ Basic Functions: + - * / numer denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: QuotientField(S) is the
++ category of fractions of an Integral Domain S.
QuotientFieldCategory(S: IntegralDomain): Category ==
  Join(Field, Algebra S, RetractableTo S, FullyEvalableOver S,
         DifferentialExtension S, FullyLinearlyExplicitRingOver S,
           Patternable S, FullyPatternMatchable S) with
    /     : (S, S) -> %
       ++ d1 / d2 returns the fraction d1 divided by d2.
    numer  : % -> S
       ++ numer(x) returns the numerator of the fraction x.
    denom  : % -> S
       ++ denom(x) returns the denominator of the fraction x.
    numerator : % -> %
       ++ numerator(x) is the numerator of the fraction x converted to %.
    denominator : % -> %
       ++ denominator(x) is the denominator of the fraction x converted to %.
    if S has StepThrough then StepThrough
    if S has RetractableTo Integer then
             RetractableTo Integer
             RetractableTo Fraction Integer
    if S has OrderedSet then OrderedSet
    if S has OrderedIntegralDomain then OrderedIntegralDomain
    if S has RealConstant then RealConstant
    if S has ConvertibleTo InputForm then ConvertibleTo InputForm
    if S has CharacteristicZero then CharacteristicZero
    if S has CharacteristicNonZero then CharacteristicNonZero
    if S has RetractableTo Symbol then RetractableTo Symbol
    if S has EuclideanDomain then
      wholePart: % -> S
        ++ wholePart(x) returns the whole part of the fraction x
        ++ i.e. the truncated quotient of the numerator by the denominator.
      fractionPart: % -> %
        ++ fractionPart(x) returns the fractional part of x.
        ++ x = wholePart(x) + fractionPart(x)
    if S has IntegerNumberSystem then
      random: () -> %
        ++ random() returns a random fraction.
      ceiling : % -> S
        ++ ceiling(x) returns the smallest integral element above x.
      floor: % -> S
        ++ floor(x) returns the largest integral element below x.
    if S has PolynomialFactorizationExplicit then
      PolynomialFactorizationExplicit

 add
    import MatrixCommonDenominator(S, %)
    numerator(x) == numer(x)::%
    denominator(x) == denom(x) ::%

    if S has StepThrough then
       init() == init()$S / 1$S

       nextItem(n) ==
         m:= nextItem(numer(n))
         m case "failed" =>
           error "We seem to have a Fraction of a finite object"
         m / 1

    map(fn, x)                         == (fn numer x) / (fn denom x)
    reducedSystem(m:Matrix %):Matrix S == clearDenominator m
    characteristic == characteristic$S

    differentiate(x:%, deriv:S -> S) ==
        n := numer x
        d := denom x
        (deriv n * d - n * deriv d) / (d**2)

    if S has ConvertibleTo InputForm then
      convert(x:%):InputForm == (convert numer x) / (convert denom x)

    if S has RealConstant then
      convert(x:%):Float == (convert numer x) / (convert denom x)
      convert(x:%):DoubleFloat == (convert numer x) / (convert denom x)

    -- Note that being a Join(OrderedSet,IntegralDomain) is not the same 
    -- as being an OrderedIntegralDomain.
    if S has OrderedIntegralDomain then
       if S has canonicalUnitNormal then
           x:% < y:% ==
             (numer x  * denom y) < (numer y * denom x)
         else
           x:% < y:% ==
             if denom(x) < 0 then (x,y):=(y,x)
             if denom(y) < 0 then (x,y):=(y,x)
             (numer x  * denom y) < (numer y * denom x)
    else if S has OrderedSet then
       x:% < y:% ==
         (numer x  * denom y) < (numer y * denom x)

    if (S has EuclideanDomain) then
      fractionPart x == x - (wholePart(x)::%)

    if S has RetractableTo Symbol then
      coerce(s:Symbol):%  == s::S::%
      retract(x:%):Symbol == retract(retract(x)@S)

      retractIfCan(x:%):Union(Symbol, "failed") ==
        (r := retractIfCan(x)@Union(S,"failed")) case "failed" =>"failed"
        retractIfCan(r::S)

    if (S has ConvertibleTo Pattern Integer) then
      convert(x:%):Pattern(Integer)==(convert numer x)/(convert denom x)

      if (S has PatternMatchable Integer) then
        patternMatch(x:%, p:Pattern Integer,
         l:PatternMatchResult(Integer, %)) ==
           patternMatch(x, p,
                     l)$PatternMatchQuotientFieldCategory(Integer, S, %)

    if (S has ConvertibleTo Pattern Float) then
      convert(x:%):Pattern(Float) == (convert numer x)/(convert denom x)

      if (S has PatternMatchable Float) then
        patternMatch(x:%, p:Pattern Float,
         l:PatternMatchResult(Float, %)) ==
           patternMatch(x, p,
                       l)$PatternMatchQuotientFieldCategory(Float, S, %)

    if S has RetractableTo Integer then
      coerce(x:Fraction Integer):% == numer(x)::% / denom(x)::%

      if not(S is Integer) then
        retract(x:%):Integer == retract(retract(x)@S)

        retractIfCan(x:%):Union(Integer, "failed") ==
          (u := retractIfCan(x)@Union(S, "failed")) case "failed" =>
            "failed"
          retractIfCan(u::S)

    if S has IntegerNumberSystem then
      random():% ==
        d : S
        while zero?(d:=random()$S) repeat d
        random()$S / d

    reducedSystem(m:Matrix %, v:Vector %):
      Record(mat:Matrix S, vec:Vector S) ==
        n := reducedSystem(horizConcat(v::Matrix(%), m))@Matrix(S)
        [subMatrix(n, minRowIndex n, maxRowIndex n, 1 + minColIndex n,
                                maxColIndex n), column(n, minColIndex n)]

@


\section{package QFCAT2 QuotientFieldCategoryFunctions2}

<<package QFCAT2 QuotientFieldCategoryFunctions2>>=
)abbrev package QFCAT2 QuotientFieldCategoryFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ This package extends a function between integral domains
++ to a mapping between their quotient fields.
QuotientFieldCategoryFunctions2(A, B, R, S): Exports == Impl where
  A, B: IntegralDomain
  R   : QuotientFieldCategory(A)
  S   : QuotientFieldCategory(B)

  Exports ==> with
    map: (A -> B, R) -> S
      ++ map(func,frac) applies the function func to the numerator
      ++ and denominator of frac.

  Impl ==> add
    map(f, r) == f(numer r) / f(denom r)

@
\section{domain FRAC Fraction}
<<domain FRAC Fraction>>=
)abbrev domain FRAC Fraction
++ Author:
++ Date Created:
++ Date Last Updated: 12 February 1992
++ Basic Functions: Field, numer, denom
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: fraction, localization
++ References:
++ Description: Fraction takes an IntegralDomain S and produces
++ the domain of Fractions with numerators and denominators from S.
++ If S is also a GcdDomain, then gcd's between numerator and
++ denominator will be cancelled during all operations.
Fraction(S: IntegralDomain): QuotientFieldCategory S with 
       if S has IntegerNumberSystem and S has OpenMath then OpenMath
       if S has canonical and S has GcdDomain and S has canonicalUnitNormal
           then canonical
            ++ \spad{canonical} means that equal elements are in fact identical.
  == LocalAlgebra(S, S, S) add
    Rep:= Record(num:S, den:S)
    coerce(d:S):% == [d,1]
    zero?(x:%) == zero? x.num


    if S has GcdDomain and S has canonicalUnitNormal then
      retract(x:%):S ==
        one?(x.den) => x.num
        error "Denominator not equal to 1"

      retractIfCan(x:%):Union(S, "failed") ==
        one?(x.den) => x.num
        "failed"
    else
      retract(x:%):S ==
        (a:= x.num exquo x.den) case "failed" =>
           error "Denominator not equal to 1"
        a
      retractIfCan(x:%):Union(S,"failed") == x.num exquo x.den

    if S has EuclideanDomain then
      wholePart x ==
        one?(x.den) => x.num
        x.num quo x.den

    if S has IntegerNumberSystem then

      floor x ==
        one?(x.den) => x.num
        x < 0 => -ceiling(-x)
        wholePart x

      ceiling x ==
        one?(x.den) => x.num
        x < 0 => -floor(-x)
        1 + wholePart x

      if S has OpenMath then
        -- TODO: somwhere this file does something which redefines the division
        -- operator. Doh!

        writeOMFrac(dev: OpenMathDevice, x: %): Void ==
          OMputApp(dev)
          OMputSymbol(dev, "nums1", "rational")
          OMwrite(dev, x.num, false)
          OMwrite(dev, x.den, false)
          OMputEndApp(dev)

        OMwrite(x: %): String ==
          s: String := ""
          sp := OM_-STRINGTOSTRINGPTR(s)$Lisp
          dev: OpenMathDevice := _
		  	OMopenString(sp pretend String, OMencodingXML())
          OMputObject(dev)
          writeOMFrac(dev, x)
          OMputEndObject(dev)
          OMclose(dev)
          s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String
          s

        OMwrite(x: %, wholeObj: Boolean): String ==
          s: String := ""
          sp := OM_-STRINGTOSTRINGPTR(s)$Lisp
          dev: OpenMathDevice := _
		  	OMopenString(sp pretend String, OMencodingXML())
          if wholeObj then
            OMputObject(dev)
          writeOMFrac(dev, x)
          if wholeObj then
            OMputEndObject(dev)
          OMclose(dev)
          s := OM_-STRINGPTRTOSTRING(sp)$Lisp pretend String
          s

        OMwrite(dev: OpenMathDevice, x: %): Void ==
          OMputObject(dev)
          writeOMFrac(dev, x)
          OMputEndObject(dev)

        OMwrite(dev: OpenMathDevice, x: %, wholeObj: Boolean): Void ==
          if wholeObj then
            OMputObject(dev)
          writeOMFrac(dev, x)
          if wholeObj then
            OMputEndObject(dev)

    if S has GcdDomain then
      cancelGcd: % -> S
      normalize: % -> %

      normalize x ==
        zero?(x.num) => 0
        one?(x.den) => x
        uca := unitNormal(x.den)
        zero?(x.den := uca.canonical) => error "division by zero"
        x.num := x.num * uca.associate
        x

      recip x ==
        zero?(x.num) => "failed"
        normalize [x.den, x.num]

      cancelGcd x ==
        one?(x.den) => x.den
        d := gcd(x.num, x.den)
        xn := x.num exquo d
        xn case "failed" =>
          error "gcd not gcd in QF cancelGcd (numerator)"
        xd := x.den exquo d
        xd case "failed" =>
          error "gcd not gcd in QF cancelGcd (denominator)"
        x.num := xn :: S
        x.den := xd :: S
        d

      nn:S / dd:S ==
        zero? dd => error "division by zero"
        cancelGcd(z := [nn, dd])
        normalize z

      x + y  ==
        zero? y => x
        zero? x => y
        z := [x.den,y.den]
        d := cancelGcd z
        g := [z.den * x.num + z.num * y.num, d]
        cancelGcd g
        g.den := g.den * z.num * z.den
        normalize g

      -- We can not rely on the defaulting mechanism
      -- to supply a definition for -, even though this
      -- definition would do, for thefollowing reasons:
      --  1) The user could have defined a subtraction
      --     in Localize, which would not work for
      --     QuotientField;
      --  2) even if he doesn't, the system currently
      --     places a default definition in Localize,
      --     which uses Localize's +, which does not
      --     cancel gcds
      x - y  ==
        zero? y => x
        z := [x.den, y.den]
        d := cancelGcd z
        g := [z.den * x.num - z.num * y.num, d]
        cancelGcd g
        g.den := g.den * z.num * z.den
        normalize g

      x:% * y:%  ==
        zero? x or zero? y => 0
        one? x => y
        one? y => x
        (x, y) := ([x.num, y.den], [y.num, x.den])
        cancelGcd x; cancelGcd y;
        normalize [x.num * y.num, x.den * y.den]

      n:Integer * x:% ==
        y := [n::S, x.den]
        cancelGcd y
        normalize [x.num * y.num, y.den]

      nn:S * x:% ==
        y := [nn, x.den]
        cancelGcd y
        normalize [x.num * y.num, y.den]

      differentiate(x:%, deriv:S -> S) ==
        y := [deriv(x.den), x.den]
        d := cancelGcd(y)
        y.num := deriv(x.num) * y.den - x.num * y.num
        (d, y.den) := (y.den, d)
        cancelGcd y
        y.den := y.den * d * d
        normalize y

      if S has canonicalUnitNormal then
        x = y == (x.num = y.num) and (x.den = y.den)
    --x / dd == (cancelGcd (z:=[x.num,dd*x.den]); normalize z)

        one? x == one? (x.num) and one? (x.den)
                  -- again assuming canonical nature of representation

    else
      nn:S/dd:S == if zero? dd then error "division by zero" else [nn,dd]

      recip x ==
        zero?(x.num) => "failed"
        [x.den, x.num]

    if (S has RetractableTo Fraction Integer) then
      retract(x:%):Fraction(Integer) == retract(retract(x)@S)

      retractIfCan(x:%):Union(Fraction Integer, "failed") ==
        (u := retractIfCan(x)@Union(S, "failed")) case "failed" => "failed"
        retractIfCan(u::S)

    else if (S has RetractableTo Integer) then
      retract(x:%):Fraction(Integer) ==
        retract(numer x) / retract(denom x)

      retractIfCan(x:%):Union(Fraction Integer, "failed") ==
        (n := retractIfCan numer x) case "failed" => "failed"
        (d := retractIfCan denom x) case "failed" => "failed"
        (n::Integer) / (d::Integer)

    QFP ==> SparseUnivariatePolynomial %
    DP ==> SparseUnivariatePolynomial S
    import UnivariatePolynomialCategoryFunctions2(%,QFP,S,DP)
    import UnivariatePolynomialCategoryFunctions2(S,DP,%,QFP)

    if S has GcdDomain then
       gcdPolynomial(pp,qq) ==
          zero? pp => qq
          zero? qq => pp
          zero? degree pp or zero? degree qq => 1
          denpp:="lcm"/[denom u for u in coefficients pp]
          ppD:DP:=map(retract(#1*denpp),pp)
          denqq:="lcm"/[denom u for u in coefficients qq]
          qqD:DP:=map(retract(#1*denqq),qq)
          g:=gcdPolynomial(ppD,qqD)
          zero? degree g => 1
          one? (lc:=leadingCoefficient g) => map(#1::%,g)
          map(#1 / lc,g)

    if (S has PolynomialFactorizationExplicit) then
       -- we'll let the solveLinearPolynomialEquations operator
       -- default from Field
       pp,qq: QFP
       lpp: List QFP
       import Factored SparseUnivariatePolynomial %
       if S has CharacteristicNonZero then
          if S has canonicalUnitNormal and S has GcdDomain then
             charthRoot x ==
               n:= charthRoot x.num
               n case "failed" => "failed"
               d:=charthRoot x.den
               d case "failed" => "failed"
               n/d
          else
             charthRoot x ==
               -- to find x = p-th root of n/d
               -- observe that xd is p-th root of n*d**(p-1)
               ans:=charthRoot(x.num *
                      (x.den)**(characteristic$%-1)::NonNegativeInteger)
               ans case "failed" => "failed"
               ans / x.den
          clear: List % -> List S
          clear l ==
             d:="lcm"/[x.den for x in l]
             [ x.num * (d exquo x.den)::S for x in l]
          mat: Matrix %
          conditionP mat ==
            matD: Matrix S
            matD:= matrix [ clear l for l in listOfLists mat ]
            ansD := conditionP matD
            ansD case "failed" => "failed"
            ansDD:=ansD :: Vector(S)
            [ ansDD(i)::% for i in 1..#ansDD]$Vector(%)

       factorPolynomial(pp) ==
          zero? pp => 0
          denpp:="lcm"/[denom u for u in coefficients pp]
          ppD:DP:=map(retract(#1*denpp),pp)
          ff:=factorPolynomial ppD
          den1:%:=denpp::%
          lfact:List Record(flg:Union("nil", "sqfr", "irred", "prime"),
                             fctr:QFP, xpnt:Integer)
          lfact:= [[w.flg,
                    if leadingCoefficient w.fctr =1 then map(#1::%,w.fctr)
                    else (lc:=(leadingCoefficient w.fctr)::%;
                           den1:=den1/lc**w.xpnt;
                            map(#1::%/lc,w.fctr)),
                   w.xpnt] for w in factorList ff]
          makeFR(map(#1::%/den1,unit(ff)),lfact)
       factorSquareFreePolynomial(pp) ==
          zero? pp => 0
          degree pp = 0 => makeFR(pp,empty())
          lcpp:=leadingCoefficient pp
          pp:=pp/lcpp
          denpp:="lcm"/[denom u for u in coefficients pp]
          ppD:DP:=map(retract(#1*denpp),pp)
          ff:=factorSquareFreePolynomial ppD
          den1:%:=denpp::%/lcpp
          lfact:List Record(flg:Union("nil", "sqfr", "irred", "prime"),
                             fctr:QFP, xpnt:Integer)
          lfact:= [[w.flg,
                    if leadingCoefficient w.fctr =1 then map(#1::%,w.fctr)
                    else (lc:=(leadingCoefficient w.fctr)::%;
                           den1:=den1/lc**w.xpnt;
                            map(#1::%/lc,w.fctr)),
                   w.xpnt] for w in factorList ff]
          makeFR(map(#1::%/den1,unit(ff)),lfact)

@
\section{package LPEFRAC LinearPolynomialEquationByFractions}
<<package LPEFRAC LinearPolynomialEquationByFractions>>=
)abbrev package LPEFRAC LinearPolynomialEquationByFractions
++ Author: James Davenport
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description:
++ Given a PolynomialFactorizationExplicit ring, this package
++ provides a defaulting rule for the \spad{solveLinearPolynomialEquation}
++ operation, by moving into the field of fractions, and solving it there
++ via the \spad{multiEuclidean} operation.
LinearPolynomialEquationByFractions(R:PolynomialFactorizationExplicit): with
  solveLinearPolynomialEquationByFractions: ( _
           List SparseUnivariatePolynomial R, _
           SparseUnivariatePolynomial R) ->   _
           Union(List SparseUnivariatePolynomial R, "failed")
        ++ solveLinearPolynomialEquationByFractions([f1, ..., fn], g)
        ++ (where the fi are relatively prime to each other)
        ++ returns a list of ai such that
        ++ \spad{g/prod fi = sum ai/fi}
        ++ or returns "failed" if no such exists.
 == add
  SupR ==> SparseUnivariatePolynomial R
  F ==> Fraction R
  SupF ==> SparseUnivariatePolynomial F
  import UnivariatePolynomialCategoryFunctions2(R,SupR,F,SupF)
  lp : List SupR
  pp: SupR
  pF: SupF
  pullback : SupF -> Union(SupR,"failed")
  pullback(pF) ==
    pF = 0 => 0
    c:=retractIfCan leadingCoefficient pF
    c case "failed" => "failed"
    r:=pullback reductum pF
    r case "failed" => "failed"
    monomial(c,degree pF) + r
  solveLinearPolynomialEquationByFractions(lp,pp) ==
    lpF:List SupF:=[map(#1@R::F,u) for u in lp]
    pF:SupF:=map(#1@R::F,pp)
    ans:= solveLinearPolynomialEquation(lpF,pF)$F
    ans case "failed" => "failed"
    [(vv:= pullback v;
      vv case "failed" => return "failed";
       vv)
        for v in ans]

@
\section{package FRAC2 FractionFunctions2}
<<package FRAC2 FractionFunctions2>>=
)abbrev package FRAC2 FractionFunctions2
++ Author:
++ Date Created:
++ Date Last Updated:
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ Description: This package extends a map between integral domains to
++ a map between Fractions over those domains by applying the map to the
++ numerators and denominators.
FractionFunctions2(A, B): Exports == Impl where
  A, B: IntegralDomain

  R ==> Fraction A
  S ==> Fraction B

  Exports ==> with
    map: (A -> B, R) -> S
      ++ map(func,frac) applies the function func to the numerator
      ++ and denominator of the fraction frac.

  Impl ==> add
    map(f, r) == map(f, r)$QuotientFieldCategoryFunctions2(A, B, R, S)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--Copyright (C) 2007-2010, Gabriel Dos Reis.
--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,
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@
<<*>>=
<<license>>

<<domain LO Localize>>
<<domain LA LocalAlgebra>>
<<category QFCAT QuotientFieldCategory>>
<<package QFCAT2 QuotientFieldCategoryFunctions2>>
<<domain FRAC Fraction>>
<<package LPEFRAC LinearPolynomialEquationByFractions>>
<<package FRAC2 FractionFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}