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Cylindrical Algebraic Decomposition

Edit detail for Cylindrical Algebraic Decomposition revision 1 of 1

	1
Editor: 127.0.0.1 Time: 2007/11/06 20:54:09 GMT-8
Note: copied from axiom-developer

changed:
-
**On Wednesday, October 12, 2005 5:29 PM Renaud Rioboo wrote:**

Dear Axiom fans,

In the last few years there have been a regain of interest in some Computer
Algebra users circles about Cylindrical Algebraic Decomposition (aka CAD). As
some of you may know my thesis was about CAD and I am regularly receiving
queries about CAD and the Axiom implementation I wrote.

I already privately gave sources or ran particular problems for some people
but I have never officially released my sources because I do not think they
are at production level and many work has still to be done on them.

Recently Martin Rubey thought it could be a good idea to release them and I
have of course no objection on that point nor on sharing these sources with
the community. If there is some interest on it you may include it into the
Axiom distribution with the same permissions than the rest of the software.

While I wait for my lab to give me the necessary permissions to be able to use
Axiom and export this software you may find it's sources at the unlisted url

http://rioboo.free.fr/CadPub/

I would like to add just a few words about that work which is a
straightforward implementation of the standard papers by Arnon, Collins and
McCallum which are more than 20 years old now.

My work on Cad is 15 years old and was developped for the defense of my thesis
where CAD was an sample application for real algebraic numbers
manipulations. By the time I wrote it I also needed some subresultant
calculations which were not in Axiom (Scratchpad by that time) and the
required machinery to work with real algebraic numbers.

Along the times, real algebraic numbers were included into Axiom and
subresultant calculations which are used in several places of the Axiom
Library were modified to all use Lionel Ducos's package which enables a
performance gain. While making these modifications to Axiom I tried to keep my
CAD package up-to-date with the Axiom Library and have used it as a test for
other packages. It thus should compile under recent Axiom versions and should
compute something that resembles a CAD.

Don't expect this package to be the absolute CAD program, I never found time
to further work on it in order to include many enhancements which are present
in other CAD packages. While Axiom versions evolved I had to remove what I
thought were enhancements and remove some support for generalization. As of
today it only includes one projection method which is the McCallum projection
and no reconstruction nor adjacency's are taken into account.

However it did reasonabily compare with more recent techniques such as
Rational Univariate Representation (aka RUR) for simple cases though it could
not produce results for some more complicated cases. By the time this
comparison was made Axiom had some severe memory restrictions which seem to
have disappear now. I thus think that many objections to using CAD could now
be revised.

For hard problems this package should thus not be worse than any other CAD
package except that you cannot fall into an optimzation drawback :-) There is
no optimization !

I will of course try to maintain the package to the best of my ability. One
thing I should do is include comments describing the different functions, but
even that requires going into the code and the algorithms which are quite old
for me now.

Let me know if there is some interest on it.

Best regards,

Renaud

<hr />

\begin{spad}
)ab pack CADU CylindricalAlgebraicDecompositionUtilities

CylindricalAlgebraicDecompositionUtilities(R,P) : PUB == PRIV where

-- Tese are some standard tools which are needed to compute with univariate
-- polynomials. 
--
-- A gcd basis for a set of polynomials is a set of pairwise relatively prime 
-- polynomials which all divide the original set and whose product is the
-- same than the product of the original set.
--
-- A square free basis for a list of polynomials is just a list
-- of square free polynomials which are pairwise relatively primes and have
-- the same roots than the original polynomials.

  R : GcdDomain
  P : UnivariatePolynomialCategory(R)

  PUB == with
      squareFreeBasis : List(P) -> List(P)
        ++ 
      gcdBasis : List(P) -> List(P)
        ++ decompose a list of polynomials into pairwise relatively 
        ++ prime polynomials
      gcdBasisAdd : (P,List(P)) -> List(P)
        ++ add one polynomial to list of pairwise relatively prime polynomials

  PRIV == add

     squareFreeBasis(lpols) ==
       lpols = [] => []
       pol := first(lpols)
       sqpol := unitCanonical(squareFreePart(pol))
       gcdBasis(cons(sqpol,squareFreeBasis(rest(lpols))))
         
     gcdBasisAdd(p,lpols) ==
       (degree(p) = 0) => lpols
       null lpols => [unitCanonical p]
       p1 := first(lpols)
       g := gcd(p,p1)
       (degree(g) = 0) => cons(p1,gcdBasisAdd(p,rest lpols))
       p := (p exquo g)::P
       p1 := (p1 exquo g)::P
       basis := gcdBasisAdd(p,rest(lpols))
       if degree(p1) > 0 then basis := cons(p1,basis)
       gcdBasisAdd(g,basis)
       

     gcdBasis(lpols) ==
       (#lpols <= 1) => lpols
       basis := gcdBasis(rest lpols)
       gcdBasisAdd(first(lpols),basis)
\end{spad}

\begin{spad}
)ab domain SCELL SimpleCell

-- This domain is made to work with one-dimensional semi-algebraic cells
-- ie either an algebraic point, either an interval. The point is specified as 
-- specification of an algebraic value.

SimpleCell(TheField,ThePols) : PUB == PRIV where
  TheField : RealClosedField
  ThePols : UnivariatePolynomialCategory(TheField)
  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger

--  VARS ==> VariationsPackage(TheField,ThePols)
  VARS ==> RealPolynomialUtilitiesPackage(TheField,ThePols)
  LF ==> List(TheField)

  PUB == CoercibleTo(O) with

     allSimpleCells : (ThePols,Symbol) -> List %
     allSimpleCells : (List(ThePols),Symbol) -> List %
     hasDimension? : % -> B
     samplePoint : % -> TheField
     stablePol : % -> ThePols
     variableOf : % -> Symbol
     separe : (LF,TheField,TheField) -> LF
     pointToCell : (TheField,B,Symbol) -> %

  PRIV == add

     Rep := Record(samplePoint:TheField,
                   hasDim:B,
                   varOf:Symbol)

     samplePoint(c) == c.samplePoint

     stablePol(c) == error "Prout"

     hasDimension?(c) == c.hasDim

     variableOf(c) == c.varOf

     coerce(c:%):O ==
       o : O := ((c.varOf)::O) = ((c.samplePoint)::O)
       brace [o,(c.hasDim)::O]

     separe(liste,gauche,droite) ==
       milieu : TheField := (gauche + droite) / (2::TheField)
       liste = [] => [milieu]
       #liste = 1 => [gauche,first(liste),droite]
       nbe := first(liste)
       lg :List(TheField) := []
       ld :List(TheField) := rest(liste)
       sg := sign(milieu-nbe)
       while sg > 0 repeat
         lg := cons(nbe,lg)
         ld = [] => return(separe(reverse(lg),gauche,milieu))
         nbe := first(ld)
         sg := sign(milieu-nbe)
         ld := rest(ld)
       sg < 0 =>
         append(separe(reverse(lg),gauche,milieu),
                rest(separe(cons(nbe,ld),milieu,droite)))
       newDroite := (gauche+milieu)/(2::TheField)
       null lg =>
           newGauche := (milieu+droite)/(2::TheField)
           while newGauche >= first(ld) repeat
             newGauche := (milieu+newGauche)/(2::TheField)
           append([gauche,milieu],separe(ld,newGauche,droite))
       while newDroite <= first(lg) repeat
         newDroite := (newDroite+milieu)/(2::TheField)
       newGauche := (milieu+droite)/(2::TheField)
       null ld => append(separe(reverse(lg),gauche,newDroite),[milieu,droite])
       while newGauche >= first(ld) repeat
         newGauche := (milieu+newGauche)/(2::TheField)
       append(separe(reverse(lg),gauche,newDroite),
              cons(milieu,separe(ld,newGauche,droite)))


     pointToCell(sp,hasDim?,varName) ==
       [sp,hasDim?,varName]$Rep

     allSimpleCells(p:ThePols,var:Symbol) ==
       allSimpleCells([p],var)

     PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols)
     allSimpleCells(lp:List(ThePols),var:Symbol) ==
       lp1 := gcdBasis(lp)$PACK
       null(lp1) => [pointToCell(0,true,var)]
       b := ("max" / [ boundOfCauchy(p)$VARS for p in lp1 ])::TheField
       l := "append" / [allRootsOf(makeSUP(unitCanonical(p))) for p in lp1]
       l := sort(l)
       l1 := separe(l,-b,b)
       res : List(%) := [pointToCell(first(l1),true,var)]
       l1 := rest(l1)
       while not(null(l1)) repeat
         res := cons(pointToCell(first(l1),false,var),res)
         l1 := rest(l1)
         l1 = [] => return(error "Liste vide")
         res := cons(pointToCell(first(l1),true,var),res)
         l1 := rest(l1)
       reverse! res
\end{spad}

\begin{spad}
)ab domain CELL Cell

Cell(TheField) : PUB == PRIV where
  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)

  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger
  BUP         ==> SparseUnivariatePolynomial(TheField)
  SCELL       ==> SimpleCell(TheField,BUP)


  PUB == CoercibleTo(O) with

     samplePoint : % -> List(TheField)
     dimension : % -> N
     hasDimension? :  (%,Symbol) -> B
     makeCell : List(SCELL) -> %
     makeCell : (SCELL,%) -> %
     mainVariableOf : % -> Symbol
     variablesOf : % -> List Symbol
     projection : % -> Union(%,"failed")
     


  PRIV == add

    Rep := List(SCELL)

    coerce(c:%):O == 
      paren [sc::O for sc in c]

    projection(cell) ==
      null cell => error "projection: should not appear"
      r := rest(cell)
      null r => "failed"
      r

    makeCell(l:List(SCELL)) == l

    makeCell(scell,toAdd) == cons(scell,toAdd)

    mainVariableOf(cell) == 
      null(cell) => 
        error "Should not appear"
      variableOf(first(cell))

    variablesOf(cell) ==
      null(cell) => []
      cons(mainVariableOf(cell),variablesOf(rest(cell)::%))

    dimension(cell) ==
      null(cell) => 0
      hasDimension?(first(cell)) => 1+dimension(rest(cell))
      dimension(rest(cell))

    hasDimension?(cell,var) ==
      null(cell) => 
        error "Should not appear"
      sc : SCELL := first(cell)
      v := variableOf(sc)
      v = var => hasDimension?(sc)
      v < var => false
      v > var => true
      error "Caca Prout"

    samplePoint(cell) ==
      null(cell) => []
      cons(samplePoint(first(cell)),samplePoint(rest(cell)))
\end{spad}

\begin{spad}
)ab pack CAD CylindricalAlgebraicDecompositionPackage

CylindricalAlgebraicDecompositionPackage(TheField) : PUB == PRIV where

  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)
  P ==> ThePols
  BUP ==> SparseUnivariatePolynomial(TheField)
  RUP ==> SparseUnivariatePolynomial(ThePols)

  Z           ==> Integer
  N           ==> NonNegativeInteger

  CELL ==> Cell(TheField)
  SCELL ==> SimpleCell(TheField,BUP)

  PUB == with

      cylindricalDecomposition: List P ->        List CELL
      cylindricalDecomposition: (List(P),List(Symbol)) ->        List CELL
      projectionSet: (List RUP)                ->    List P
      coefficientSet: RUP                ->    List P
      discriminantSet : List RUP -> List(P)
      resultantSet :  List RUP -> List P 
      principalSubResultantSet : (RUP,RUP) -> List P
      specialise : (List(ThePols),CELL) -> List(BUP)

  PRIV == add

     cylindricalDecomposition(lpols) ==
       lv : List(Symbol) := []
       for pol in lpols repeat
         ground?(pol) => "next pol"
         lv := removeDuplicates(append(variables(pol),lv))
       lv := reverse(sort(lv))
       cylindricalDecomposition(lpols,lv)

     cylindricalDecomposition(lpols,lvars) ==
       lvars = [] => error("CAD: cylindricalDecomposition: empty list of vars")
       mv := first(lvars)
       lv := rest(lvars)
       lv = [] =>
         lp1 := [ univariate(pol) for pol in lpols ]
         scells := allSimpleCells(lp1,mv)$SCELL
         [ makeCell([scell]) for scell in scells ]
       lpols1 := projectionSet [univariate(pol,mv) for pol in lpols]
       previousCad := cylindricalDecomposition(lpols1,lv)
       res : List(CELL) := []
       for cell in previousCad repeat
         lspec := specialise(lpols,cell)
         scells := allSimpleCells(lspec,mv)
         res := append(res,[makeCell(scell,cell) for scell in scells])
       res


     PACK1 ==> CylindricalAlgebraicDecompositionUtilities(ThePols,RUP)
     PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,BUP)

     specialise(lpols,cell) ==
       lpols = [] => error("CAD: specialise: empty list of pols")
       sp := samplePoint(cell)
       vl := variablesOf(cell)
       res : List(BUP) := []
       for pol in lpols repeat
         p1 := univariate(eval(pol,vl,sp))
--         zero?(p1) => return(error "Bad sepcialise")
         degree(p1) = 0 => "next pol"
         res := cons(p1,res)
       res


     coefficientSet(pol) ==
       res : List(ThePols) := []
       for c in coefficients(pol) repeat
         ground?(c) => return(res)
         res := cons(c,res)
       res

     SUBRES ==> SubResultantPackage(ThePols,RUP)
     discriminantSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         v := subresultantVector(p,differentiate(p))$SUBRES
         not(zero?(degree(v.0))) => return(error "Bad discriminant")
         d : ThePols :=  leadingCoefficient(v.0)
--         d := discriminant p
         zero?(d) => return(error "Non Square Free polynomial")
         if not(ground? d) then res := cons(d,res)
       res

     principalSubResultantSet(p,q) ==
        if degree(p) < degree(q)
        then (p,q) := (q,p)
        if degree(p) = degree(q)
        then (p,q) := (q,pseudoRemainder(p, q))
        v := subresultantVector(p,q)$SUBRES
        [coefficient(v.i,i) for i in 0..(((#v)-2)::N)]

     resultantSet(lpols) ==
       res : List(ThePols) := []
       laux := lpols
       for p in lpols repeat
         laux := rest(laux)
         for q in laux repeat
           r : ThePols :=  first(principalSubResultantSet(p,q))
--           r := resultant(p,q)
           zero?(r) => return(error "Non relatively prime polynomials")
           if not(ground? r) then res := cons(r,res)
       res

     projectionSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         c := content(p)
         ground?(c) => "next p"
         res := cons(c,res)
       lp1 := [primitivePart p for p in lpols]
       f : ((RUP,RUP) -> Boolean) := (degree(#1) <= degree(#2))
       lp1 := sort(f,lp1)
       lsqfrb := squareFreeBasis(lp1)$PACK1
       lsqfrb := sort(f,lsqfrb)
       for p in lp1 repeat
         res := append(res,coefficientSet(p))
       res := append(res,discriminantSet(lsqfrb))
       append(res,resultantSet(lsqfrb))
\end{spad}

\begin{axiom}
)lib )dir .
)lib SCELL CELL CAD
Ran := RECLOS(FRAC INT)
Cad := CAD(Ran)
p1 : POLY(Ran) := x^2+y^2-1
p2 : POLY(Ran) := y-x^2
lpols : List(POLY(Ran)) := [p1,p2]
cad := cylindricalDecomposition(lpols)$Cad
precision(30)
ls := [ samplePoint cell for cell in cad]
lf := [ [relativeApprox(coord,2^(-precision()))::Float for coord in point] for point in ls]
lp := [ point(term::List(DoubleFloat))$Point(DoubleFloat) for term in lf ]
[ sign(eval(p1,variables(p1),samplePoint(cell))::Ran) for cell in cad ]
--[ sign(eval(p2,variables(p2),samplePoint(cell))::Ran) for cell in cad ]
\end{axiom}

On Wednesday, October 12, 2005 5:29 PM Renaud Rioboo wrote:

Dear Axiom fans,

In the last few years there have been a regain of interest in some Computer Algebra users circles about Cylindrical Algebraic Decomposition (aka CAD). As some of you may know my thesis was about CAD and I am regularly receiving queries about CAD and the Axiom implementation I wrote.

I already privately gave sources or ran particular problems for some people but I have never officially released my sources because I do not think they are at production level and many work has still to be done on them.

Recently Martin Rubey thought it could be a good idea to release them and I have of course no objection on that point nor on sharing these sources with the community. If there is some interest on it you may include it into the Axiom distribution with the same permissions than the rest of the software.

While I wait for my lab to give me the necessary permissions to be able to use Axiom and export this software you may find it's sources at the unlisted url

http://rioboo.free.fr/CadPub/

I would like to add just a few words about that work which is a straightforward implementation of the standard papers by Arnon, Collins and McCallum? which are more than 20 years old now.

My work on Cad is 15 years old and was developped for the defense of my thesis where CAD was an sample application for real algebraic numbers manipulations. By the time I wrote it I also needed some subresultant calculations which were not in Axiom (Scratchpad by that time) and the required machinery to work with real algebraic numbers.

Along the times, real algebraic numbers were included into Axiom and subresultant calculations which are used in several places of the Axiom Library were modified to all use Lionel Ducos's package which enables a performance gain. While making these modifications to Axiom I tried to keep my CAD package up-to-date with the Axiom Library and have used it as a test for other packages. It thus should compile under recent Axiom versions and should compute something that resembles a CAD.

Don't expect this package to be the absolute CAD program, I never found time to further work on it in order to include many enhancements which are present in other CAD packages. While Axiom versions evolved I had to remove what I thought were enhancements and remove some support for generalization. As of today it only includes one projection method which is the McCallum? projection and no reconstruction nor adjacency's are taken into account.

However it did reasonabily compare with more recent techniques such as Rational Univariate Representation (aka RUR) for simple cases though it could not produce results for some more complicated cases. By the time this comparison was made Axiom had some severe memory restrictions which seem to have disappear now. I thus think that many objections to using CAD could now be revised.

For hard problems this package should thus not be worse than any other CAD package except that you cannot fall into an optimzation drawback :-) There is no optimization !

I will of course try to maintain the package to the best of my ability. One thing I should do is include comments describing the different functions, but even that requires going into the code and the algorithms which are quite old for me now.

Let me know if there is some interest on it.

Best regards,

Renaud

spad

)ab pack CADU CylindricalAlgebraicDecompositionUtilities

CylindricalAlgebraicDecompositionUtilities(R,P) : PUB == PRIV where

-- Tese are some standard tools which are needed to compute with univariate
-- polynomials. 
--
-- A gcd basis for a set of polynomials is a set of pairwise relatively prime 
-- polynomials which all divide the original set and whose product is the
-- same than the product of the original set.
--
-- A square free basis for a list of polynomials is just a list
-- of square free polynomials which are pairwise relatively primes and have
-- the same roots than the original polynomials.

  R : GcdDomain
  P : UnivariatePolynomialCategory(R)

  PUB == with
      squareFreeBasis : List(P) -> List(P)
        ++ 
      gcdBasis : List(P) -> List(P)
        ++ decompose a list of polynomials into pairwise relatively 
        ++ prime polynomials
      gcdBasisAdd : (P,List(P)) -> List(P)
        ++ add one polynomial to list of pairwise relatively prime polynomials

  PRIV == add

     squareFreeBasis(lpols) ==
       lpols = [] => []
       pol := first(lpols)
       sqpol := unitCanonical(squareFreePart(pol))
       gcdBasis(cons(sqpol,squareFreeBasis(rest(lpols))))

     gcdBasisAdd(p,lpols) ==
       (degree(p) = 0) => lpols
       null lpols => [unitCanonical p]
       p1 := first(lpols)
       g := gcd(p,p1)
       (degree(g) = 0) => cons(p1,gcdBasisAdd(p,rest lpols))
       p := (p exquo g)::P
       p1 := (p1 exquo g)::P
       basis := gcdBasisAdd(p,rest(lpols))
       if degree(p1) > 0 then basis := cons(p1,basis)
       gcdBasisAdd(g,basis)

     gcdBasis(lpols) ==
       (#lpols <= 1) => lpols
       basis := gcdBasis(rest lpols)
       gcdBasisAdd(first(lpols),basis)

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/4986109030324487666-25px001.spad using 
      old system compiler.
   CADU abbreviates package CylindricalAlgebraicDecompositionUtilities 
------------------------------------------------------------------------
   initializing NRLIB CADU for CylindricalAlgebraicDecompositionUtilities 
   compiling into NRLIB CADU 
   compiling exported squareFreeBasis : List P -> List P
Time: 0.18 SEC.

   compiling exported gcdBasisAdd : (P,List P) -> List P
Time: 0.02 SEC.

   compiling exported gcdBasis : List P -> List P
Time: 0.04 SEC.

(time taken in buildFunctor:  6)

;;;     ***       |CylindricalAlgebraicDecompositionUtilities| REDEFINED

;;;     ***       |CylindricalAlgebraicDecompositionUtilities| REDEFINED
Time: 0.06 SEC.

   Cumulative Statistics for Constructor CylindricalAlgebraicDecompositionUtilities
      Time: 0.30 seconds

   finalizing NRLIB CADU 
   Processing CylindricalAlgebraicDecompositionUtilities for Browser database:
--------(squareFreeBasis ((List P) (List P)))---------
--------(gcdBasis ((List P) (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasis ((List P) (List P)))): Improper first word in comments: decompose
"decompose a list of polynomials into pairwise relatively prime polynomials"
--------(gcdBasisAdd ((List P) P (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasisAdd ((List P) P (List P)))): Improper first word in comments: add
"add one polynomial to list of pairwise relatively prime polynomials"
--->-->CylindricalAlgebraicDecompositionUtilities(constructor): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionUtilities(): Missing Description
------------------------------------------------------------------------
   CylindricalAlgebraicDecompositionUtilities is now explicitly exposed
      in frame initial 
   CylindricalAlgebraicDecompositionUtilities will be automatically 
      loaded when needed from /var/zope2/var/LatexWiki/CADU.NRLIB/code

spad )ab domain SCELL SimpleCell

This domain is made to work with one-dimensional semi-algebraic cells -- ie either an algebraic point, either an interval. The point is specified as -- specification of an algebraic value.
SimpleCell(TheField,ThePols) : PUB == PRIV where TheField : RealClosedField ThePols : UnivariatePolynomialCategory(TheField) O ==> OutputForm B ==> Boolean Z ==> Integer N ==> NonNegativeInteger
-- VARS ==> VariationsPackage(TheField,ThePols) VARS ==> RealPolynomialUtilitiesPackage(TheField,ThePols) LF ==> List(TheField)
PUB == CoercibleTo(O) with
allSimpleCells : (ThePols,Symbol) -> List % allSimpleCells : (List(ThePols),Symbol) -> List % hasDimension? : % -> B samplePoint : % -> TheField stablePol : % -> ThePols variableOf : % -> Symbol separe : (LF,TheField,TheField) -> LF pointToCell : (TheField,B,Symbol) -> %
PRIV == add
Rep := Record(samplePoint:TheField, hasDim:B, varOf:Symbol)
samplePoint(c) == c.samplePoint
stablePol(c) == error "Prout"
hasDimension?(c) == c.hasDim
variableOf(c) == c.varOf
coerce(c:%):O == o : O := ((c.varOf)::O) = ((c.samplePoint)::O) brace [o,(c.hasDim)::O]
separe(liste,gauche,droite) == milieu : TheField := (gauche + droite) / (2::TheField) liste = [] => [milieu] #liste = 1 => [gauche,first(liste),droite] nbe := first(liste) lg :List(TheField) := [] ld :List(TheField) := rest(liste) sg := sign(milieu-nbe) while sg > 0 repeat lg := cons(nbe,lg) ld = [] => return(separe(reverse(lg),gauche,milieu)) nbe := first(ld) sg := sign(milieu-nbe) ld := rest(ld) sg < 0 => append(separe(reverse(lg),gauche,milieu), rest(separe(cons(nbe,ld),milieu,droite))) newDroite := (gauche+milieu)/(2::TheField) null lg => newGauche := (milieu+droite)/(2::TheField) while newGauche >= first(ld) repeat newGauche := (milieu+newGauche)/(2::TheField) append([gauche,milieu],separe(ld,newGauche,droite)) while newDroite <= first(lg) repeat newDroite := (newDroite+milieu)/(2::TheField) newGauche := (milieu+droite)/(2::TheField) null ld => append(separe(reverse(lg),gauche,newDroite),[milieu,droite]) while newGauche >= first(ld) repeat newGauche := (milieu+newGauche)/(2::TheField) append(separe(reverse(lg),gauche,newDroite), cons(milieu,separe(ld,newGauche,droite)))
pointToCell(sp,hasDim?,varName) == [sp,hasDim?,varName]$Rep
allSimpleCells(p:ThePols,var:Symbol) == allSimpleCells([p],var)
PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols) allSimpleCells(lp:List(ThePols),var:Symbol) == lp1 := gcdBasis(lp)$PACK null(lp1) => [pointToCell(0,true,var)] b := ("max" / [ boundOfCauchy(p)$VARS for p in lp1 ])::TheField l := "append" / [allRootsOf(makeSUP(unitCanonical(p))) for p in lp1] l := sort(l) l1 := separe(l,-b,b) res : List(%) := [pointToCell(first(l1),true,var)] l1 := rest(l1) while not(null(l1)) repeat res := cons(pointToCell(first(l1),false,var),res) l1 := rest(l1) l1 = [] => return(error "Liste vide") res := cons(pointToCell(first(l1),true,var),res) l1 := rest(l1) reverse! res

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/9165941727097331838-25px002.spad using 
      old system compiler.
   SCELL abbreviates domain SimpleCell 
   processing macro definition O ==> OutputForm 
   processing macro definition B ==> Boolean 
   processing macro definition Z ==> Integer 
   processing macro definition N ==> NonNegativeInteger 
   processing macro definition VARS ==> RealPolynomialUtilitiesPackage(TheField,ThePols) 
   processing macro definition LF ==> List TheField 
------------------------------------------------------------------------
   initializing NRLIB SCELL for SimpleCell 
   compiling into NRLIB SCELL 
   compiling exported samplePoint : $ -> TheField
      SCELL;samplePoint;$TheField;1 is replaced by QVELTc0 
Time: 0.03 SEC.

   compiling exported stablePol : $ -> ThePols
      SCELL;stablePol;$ThePols;2 is replaced by errorProut 
Time: 0 SEC.

   compiling exported hasDimension? : $ -> Boolean
      SCELL;hasDimension?;$B;3 is replaced by QVELTc1 
Time: 0 SEC.

   compiling exported variableOf : $ -> Symbol
      SCELL;variableOf;$S;4 is replaced by QVELTc2 
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0.11 SEC.

   compiling exported separe : (List TheField,TheField,TheField) -> List TheField
Time: 0.05 SEC.

   compiling exported pointToCell : (TheField,Boolean,Symbol) -> $
      SCELL;pointToCell;TheFieldBS$;7 is replaced by VECTOR 
Time: 0 SEC.

   compiling exported allSimpleCells : (ThePols,Symbol) -> List $
Time: 0.02 SEC.

   processing macro definition PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols) 
   compiling exported allSimpleCells : (List ThePols,Symbol) -> List $
Time: 0.12 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SimpleCell| REDEFINED

;;;     ***       |SimpleCell| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] allSimpleCells: not known that (Ring) is of mode (CATEGORY domain (SIGNATURE allSimpleCells ((List $) ThePols (Symbol))) (SIGNATURE allSimpleCells ((List $) (List ThePols) (Symbol))) (SIGNATURE hasDimension? ((Boolean) $)) (SIGNATURE samplePoint (TheField $)) (SIGNATURE stablePol (ThePols $)) (SIGNATURE variableOf ((Symbol) $)) (SIGNATURE separe ((List TheField) (List TheField) TheField TheField)) (SIGNATURE pointToCell ($ TheField (Boolean) (Symbol))))

   Cumulative Statistics for Constructor SimpleCell
      Time: 0.33 seconds

   finalizing NRLIB SCELL 
   Processing SimpleCell for Browser database:
--->-->SimpleCell((allSimpleCells ((List %) ThePols (Symbol)))): Not documented!!!!
--->-->SimpleCell((allSimpleCells ((List %) (List ThePols) (Symbol)))): Not documented!!!!
--->-->SimpleCell((hasDimension? (B %))): Not documented!!!!
--->-->SimpleCell((samplePoint (TheField %))): Not documented!!!!
--->-->SimpleCell((stablePol (ThePols %))): Not documented!!!!
--->-->SimpleCell((variableOf ((Symbol) %))): Not documented!!!!
--->-->SimpleCell((separe (LF LF TheField TheField))): Not documented!!!!
--->-->SimpleCell((pointToCell (% TheField B (Symbol)))): Not documented!!!!
--->-->SimpleCell(constructor): Not documented!!!!
--->-->SimpleCell(): Missing Description
------------------------------------------------------------------------
   SimpleCell is now explicitly exposed in frame initial 
   SimpleCell will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/SCELL.NRLIB/code

spad

)ab domain CELL Cell

Cell(TheField) : PUB == PRIV where
  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)

  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger
  BUP         ==> SparseUnivariatePolynomial(TheField)
  SCELL       ==> SimpleCell(TheField,BUP)

  PUB == CoercibleTo(O) with

     samplePoint : % -> List(TheField)
     dimension : % -> N
     hasDimension? :  (%,Symbol) -> B
     makeCell : List(SCELL) -> %
     makeCell : (SCELL,%) -> %
     mainVariableOf : % -> Symbol
     variablesOf : % -> List Symbol
     projection : % -> Union(%,"failed")

  PRIV == add

    Rep := List(SCELL)

    coerce(c:%):O == 
      paren [sc::O for sc in c]

    projection(cell) ==
      null cell => error "projection: should not appear"
      r := rest(cell)
      null r => "failed"
      r

    makeCell(l:List(SCELL)) == l

    makeCell(scell,toAdd) == cons(scell,toAdd)

    mainVariableOf(cell) == 
      null(cell) => 
        error "Should not appear"
      variableOf(first(cell))

    variablesOf(cell) ==
      null(cell) => []
      cons(mainVariableOf(cell),variablesOf(rest(cell)::%))

    dimension(cell) ==
      null(cell) => 0
      hasDimension?(first(cell)) => 1+dimension(rest(cell))
      dimension(rest(cell))

    hasDimension?(cell,var) ==
      null(cell) => 
        error "Should not appear"
      sc : SCELL := first(cell)
      v := variableOf(sc)
      v = var => hasDimension?(sc)
      v < var => false
      v > var => true
      error "Caca Prout"

    samplePoint(cell) ==
      null(cell) => []
      cons(samplePoint(first(cell)),samplePoint(rest(cell)))

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/6660987790622940247-25px003.spad using 
      old system compiler.
   CELL abbreviates domain Cell 
   processing macro definition ThePols ==> Polynomial TheField 
   processing macro definition O ==> OutputForm 
   processing macro definition B ==> Boolean 
   processing macro definition Z ==> Integer 
   processing macro definition N ==> NonNegativeInteger 
   processing macro definition BUP ==> SparseUnivariatePolynomial TheField 
   processing macro definition SCELL ==> SimpleCell(TheField,SparseUnivariatePolynomial TheField) 
------------------------------------------------------------------------
   initializing NRLIB CELL for Cell 
   compiling into NRLIB CELL 
   compiling exported coerce : $ -> OutputForm
Time: 0.01 SEC.

   compiling exported projection : $ -> Union($,failed)
Time: 0.01 SEC.

   compiling exported makeCell : List SimpleCell(TheField,SparseUnivariatePolynomial TheField) -> $
      CELL;makeCell;L$;3 is replaced by l 
Time: 0 SEC.

   compiling exported makeCell : (SimpleCell(TheField,SparseUnivariatePolynomial TheField),$) -> $
      CELL;makeCell;Sc2$;4 is replaced by CONS 
Time: 0 SEC.

   compiling exported mainVariableOf : $ -> Symbol
Time: 0.01 SEC.

   compiling exported variablesOf : $ -> List Symbol
Time: 0 SEC.

   compiling exported dimension : $ -> NonNegativeInteger
Time: 0.01 SEC.

   compiling exported hasDimension? : ($,Symbol) -> Boolean
Time: 0.01 SEC.

   compiling exported samplePoint : $ -> List TheField
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Cell| REDEFINED

;;;     ***       |Cell| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor Cell
      Time: 0.05 seconds

   finalizing NRLIB CELL 
   Processing Cell for Browser database:
--->-->Cell((samplePoint ((List TheField) %))): Not documented!!!!
--->-->Cell((dimension (N %))): Not documented!!!!
--->-->Cell((hasDimension? (B % (Symbol)))): Not documented!!!!
--->-->Cell((makeCell (% (List SCELL)))): Not documented!!!!
--->-->Cell((makeCell (% SCELL %))): Not documented!!!!
--->-->Cell((mainVariableOf ((Symbol) %))): Not documented!!!!
--->-->Cell((variablesOf ((List (Symbol)) %))): Not documented!!!!
--->-->Cell((projection ((Union % failed) %))): Not documented!!!!
--->-->Cell(constructor): Not documented!!!!
--->-->Cell(): Missing Description
------------------------------------------------------------------------
   Cell is now explicitly exposed in frame initial 
   Cell will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/CELL.NRLIB/code

spad

)ab pack CAD CylindricalAlgebraicDecompositionPackage

CylindricalAlgebraicDecompositionPackage(TheField) : PUB == PRIV where

  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)
  P ==> ThePols
  BUP ==> SparseUnivariatePolynomial(TheField)
  RUP ==> SparseUnivariatePolynomial(ThePols)

  Z           ==> Integer
  N           ==> NonNegativeInteger

  CELL ==> Cell(TheField)
  SCELL ==> SimpleCell(TheField,BUP)

  PUB == with

      cylindricalDecomposition: List P ->        List CELL
      cylindricalDecomposition: (List(P),List(Symbol)) ->        List CELL
      projectionSet: (List RUP)                ->    List P
      coefficientSet: RUP                ->    List P
      discriminantSet : List RUP -> List(P)
      resultantSet :  List RUP -> List P 
      principalSubResultantSet : (RUP,RUP) -> List P
      specialise : (List(ThePols),CELL) -> List(BUP)

  PRIV == add

     cylindricalDecomposition(lpols) ==
       lv : List(Symbol) := []
       for pol in lpols repeat
         ground?(pol) => "next pol"
         lv := removeDuplicates(append(variables(pol),lv))
       lv := reverse(sort(lv))
       cylindricalDecomposition(lpols,lv)

     cylindricalDecomposition(lpols,lvars) ==
       lvars = [] => error("CAD: cylindricalDecomposition: empty list of vars")
       mv := first(lvars)
       lv := rest(lvars)
       lv = [] =>
         lp1 := [ univariate(pol) for pol in lpols ]
         scells := allSimpleCells(lp1,mv)$SCELL
         [ makeCell([scell]) for scell in scells ]
       lpols1 := projectionSet [univariate(pol,mv) for pol in lpols]
       previousCad := cylindricalDecomposition(lpols1,lv)
       res : List(CELL) := []
       for cell in previousCad repeat
         lspec := specialise(lpols,cell)
         scells := allSimpleCells(lspec,mv)
         res := append(res,[makeCell(scell,cell) for scell in scells])
       res

     PACK1 ==> CylindricalAlgebraicDecompositionUtilities(ThePols,RUP)
     PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,BUP)

     specialise(lpols,cell) ==
       lpols = [] => error("CAD: specialise: empty list of pols")
       sp := samplePoint(cell)
       vl := variablesOf(cell)
       res : List(BUP) := []
       for pol in lpols repeat
         p1 := univariate(eval(pol,vl,sp))
--         zero?(p1) => return(error "Bad sepcialise")
         degree(p1) = 0 => "next pol"
         res := cons(p1,res)
       res

     coefficientSet(pol) ==
       res : List(ThePols) := []
       for c in coefficients(pol) repeat
         ground?(c) => return(res)
         res := cons(c,res)
       res

     SUBRES ==> SubResultantPackage(ThePols,RUP)
     discriminantSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         v := subresultantVector(p,differentiate(p))$SUBRES
         not(zero?(degree(v.0))) => return(error "Bad discriminant")
         d : ThePols :=  leadingCoefficient(v.0)
--         d := discriminant p
         zero?(d) => return(error "Non Square Free polynomial")
         if not(ground? d) then res := cons(d,res)
       res

     principalSubResultantSet(p,q) ==
        if degree(p) < degree(q)
        then (p,q) := (q,p)
        if degree(p) = degree(q)
        then (p,q) := (q,pseudoRemainder(p, q))
        v := subresultantVector(p,q)$SUBRES
        [coefficient(v.i,i) for i in 0..(((#v)-2)::N)]

     resultantSet(lpols) ==
       res : List(ThePols) := []
       laux := lpols
       for p in lpols repeat
         laux := rest(laux)
         for q in laux repeat
           r : ThePols :=  first(principalSubResultantSet(p,q))
--           r := resultant(p,q)
           zero?(r) => return(error "Non relatively prime polynomials")
           if not(ground? r) then res := cons(r,res)
       res

     projectionSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         c := content(p)
         ground?(c) => "next p"
         res := cons(c,res)
       lp1 := [primitivePart p for p in lpols]
       f : ((RUP,RUP) -> Boolean) := (degree(#1) <= degree(#2))
       lp1 := sort(f,lp1)
       lsqfrb := squareFreeBasis(lp1)$PACK1
       lsqfrb := sort(f,lsqfrb)
       for p in lp1 repeat
         res := append(res,coefficientSet(p))
       res := append(res,discriminantSet(lsqfrb))
       append(res,resultantSet(lsqfrb))

spad

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/1235315989953270759-25px004.spad using 
      old system compiler.
   CAD abbreviates package CylindricalAlgebraicDecompositionPackage 
   processing macro definition ThePols ==> Polynomial TheField 
   processing macro definition P ==> Polynomial TheField 
   processing macro definition BUP ==> SparseUnivariatePolynomial TheField 
   processing macro definition RUP ==> SparseUnivariatePolynomial Polynomial TheField 
   processing macro definition Z ==> Integer 
   processing macro definition N ==> NonNegativeInteger 
   processing macro definition CELL ==> Cell TheField 
   processing macro definition SCELL ==> SimpleCell(TheField,SparseUnivariatePolynomial TheField) 
------------------------------------------------------------------------
   initializing NRLIB CAD for CylindricalAlgebraicDecompositionPackage 
   compiling into NRLIB CAD 
   compiling exported cylindricalDecomposition : List Polynomial TheField -> List Cell TheField
Time: 0.04 SEC.

   compiling exported cylindricalDecomposition : (List Polynomial TheField,List Symbol) -> List Cell TheField
Time: 0.46 SEC.

   processing macro definition PACK1 ==> CylindricalAlgebraicDecompositionUtilities(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   processing macro definition PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,SparseUnivariatePolynomial TheField) 
   compiling exported specialise : (List Polynomial TheField,Cell TheField) -> List SparseUnivariatePolynomial TheField
Time: 0.28 SEC.

   compiling exported coefficientSet : SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.01 SEC.

   processing macro definition SUBRES ==> SubResultantPackage(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   compiling exported discriminantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.17 SEC.

   compiling exported principalSubResultantSet : (SparseUnivariatePolynomial Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) -> List Polynomial TheField
Time: 0.17 SEC.

   compiling exported resultantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.03 SEC.

   compiling exported projectionSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.08 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |CylindricalAlgebraicDecompositionPackage| REDEFINED

;;;     ***       |CylindricalAlgebraicDecompositionPackage| REDEFINED
Time: 0 SEC.

   Warnings: 
      [1] cylindricalDecomposition:  lv has no value
      [2] cylindricalDecomposition: not known that (Ring) is of mode (CATEGORY package (SIGNATURE cylindricalDecomposition ((List (Cell TheField)) (List (Polynomial TheField)))) (SIGNATURE cylindricalDecomposition ((List (Cell TheField)) (List (Polynomial TheField)) (List (Symbol)))) (SIGNATURE projectionSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField))))) (SIGNATURE coefficientSet ((List (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField)))) (SIGNATURE discriminantSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField))))) (SIGNATURE resultantSet ((List (Polynomial TheField)) (List (SparseUnivariatePolynomial (Polynomial TheField))))) (SIGNATURE principalSubResultantSet ((List (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField)) (SparseUnivariatePolynomial (Polynomial TheField)))) (SIGNATURE specialise ((List (SparseUnivariatePolynomial TheField)) (List (Polynomial TheField)) (Cell TheField))))
      [3] specialise:  res has no value
      [4] discriminantSet:  res has no value
      [5] resultantSet:  res has no value
      [6] projectionSet:  res has no value

   Cumulative Statistics for Constructor CylindricalAlgebraicDecompositionPackage
      Time: 1.24 seconds

   finalizing NRLIB CAD 
   Processing CylindricalAlgebraicDecompositionPackage for Browser database:
--->-->CylindricalAlgebraicDecompositionPackage((cylindricalDecomposition ((List CELL) (List P)))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((cylindricalDecomposition ((List CELL) (List P) (List (Symbol))))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((projectionSet ((List P) (List RUP)))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((coefficientSet ((List P) RUP))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((discriminantSet ((List P) (List RUP)))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((resultantSet ((List P) (List RUP)))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((principalSubResultantSet ((List P) RUP RUP))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage((specialise ((List BUP) (List ThePols) CELL))): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage(constructor): Not documented!!!!
--->-->CylindricalAlgebraicDecompositionPackage(): Missing Description
------------------------------------------------------------------------
   CylindricalAlgebraicDecompositionPackage is now explicitly exposed 
      in frame initial 
   CylindricalAlgebraicDecompositionPackage will be automatically 
      loaded when needed from /var/zope2/var/LatexWiki/CAD.NRLIB/code

axiom

)lib )dir .

   XPrimitiveArray is now explicitly exposed in frame initial
   XPrimitiveArray will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/XPRIMARR.NRLIB/code
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      /var/zope2/var/LatexWiki/WORD.NRLIB/code
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      /var/zope2/var/LatexWiki/UNITS.NRLIB/code
   UnivariateFormalPowerSeriesFunctions is now explicitly exposed in
      frame initial
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      when needed from /var/zope2/var/LatexWiki/UFPS1.NRLIB/code
   UnivariateFormalPowerSeries is now explicitly exposed in frame
      initial
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      from /var/zope2/var/LatexWiki/UFPS.NRLIB/code
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      /var/zope2/var/LatexWiki/TT.NRLIB/code
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      initial
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      needed from /var/zope2/var/LatexWiki/TRMANIP2.NRLIB/code
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      /var/zope2/var/LatexWiki/TPROD2.NRLIB/code
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      /var/zope2/var/LatexWiki/TPROD.NRLIB/code
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      /var/zope2/var/LatexWiki/SYMS.NRLIB/code
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      /var/zope2/var/LatexWiki/SVT3.NRLIB/code
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      /var/zope2/var/LatexWiki/SVT1.NRLIB/code
   SparseUnivariatePolynomialExpressions is now explicitly exposed in
      frame initial
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      when needed from /var/zope2/var/LatexWiki/SUPEXPR.NRLIB/code
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      /var/zope2/var/LatexWiki/SUM.NRLIB/code
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      /var/zope2/var/LatexWiki/SUBDOM.NRLIB/code
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      /var/zope2/var/LatexWiki/STDOMAIN.NRLIB/code
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      from /var/zope2/var/LatexWiki/SFX.NRLIB/code
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      from /var/zope2/var/LatexWiki/SEGXCAT.NRLIB/code
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      /var/zope2/var/LatexWiki/SCELL.NRLIB/code
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      initial
   RationalInterpolationAlgorithms will be automatically loaded when
      needed from /var/zope2/var/LatexWiki/RINTERPA.NRLIB/code
   RationalInterpolation is now explicitly exposed in frame initial
   RationalInterpolation will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/RINTERP.NRLIB/code
   Ring is now explicitly exposed in frame initial
   Ring will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/RING.NRLIB/code
   Relations is now explicitly exposed in frame initial
   Relations will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/REL.NRLIB/code
   Reflect is now explicitly exposed in frame initial
   Reflect will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/REFL.NRLIB/code
   RecurrenceOperator is now explicitly exposed in frame initial
   RecurrenceOperator will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/RECOP.NRLIB/code
   ReadFile is now explicitly exposed in frame initial
   ReadFile will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/READFILE.NRLIB/code
   QuadraticForm is now explicitly exposed in frame initial
   QuadraticForm will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/QFORM.NRLIB/code
   Product is now explicitly exposed in frame initial
   Product will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PRODUCT.NRLIB/code
   PointedPrimeField is now explicitly exposed in frame initial
   PointedPrimeField will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PPF.NRLIB/code
   Polytope is now explicitly exposed in frame initial
   Polytope will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/POLYTOPE.NRLIB/code
   PolynomialCategory is now explicitly exposed in frame initial
   PolynomialCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/POLYCAT.NRLIB/code
   PolynomialCategory& is now explicitly exposed in frame initial
   PolynomialCategory& will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/POLYCAT-.NRLIB/code
   Poly2? is now explicitly exposed in frame initial
   Poly2? will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/POLY2?.NRLIB/code
   Poly1? is now explicitly exposed in frame initial
   Poly1? will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/POLY1?.NRLIB/code
   PositiveInteger is now explicitly exposed in frame initial
   PositiveInteger will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PI.NRLIB/code
   PolynomialCommonDenominator is now explicitly exposed in frame
      initial
   PolynomialCommonDenominator will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PCDEN.NRLIB/code
   Parity is now explicitly exposed in frame initial
   Parity will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PARITY.NRLIB/code
   OptimizationPackage is now explicitly exposed in frame initial
   OptimizationPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/OPT.NRLIB/code
   OrderedFreeMonoid is now explicitly exposed in frame initial
   OrderedFreeMonoid will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/OFMONOID.NRLIB/code
   odd is now explicitly exposed in frame initial
   odd will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/ODD.NRLIB/code
   Object is now explicitly exposed in frame initial
   Object will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/OBJ.NRLIB/code
   NonNegativeInteger2 is now explicitly exposed in frame initial
   NonNegativeInteger2 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NNI2.NRLIB/code
   NonNegativeInteger is now explicitly exposed in frame initial
   NonNegativeInteger will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NNI.NRLIB/code
   NewtonInterpolation is now explicitly exposed in frame initial
   NewtonInterpolation will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NEWTON.NRLIB/code
   NewMonoid is now explicitly exposed in frame initial
   NewMonoid will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NEWMON.NRLIB/code
   NewMonoid& is now explicitly exposed in frame initial
   NewMonoid& will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NEWMON-.NRLIB/code
   NewInteger is now explicitly exposed in frame initial
   NewInteger will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NEWINT.NRLIB/code
   Inequation is now explicitly exposed in frame initial
   Inequation will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NEQ.NRLIB/code
   Inequation is already explicitly exposed in frame initial
   Inequation will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/NE.NRLIB/code
   MySubset is now explicitly exposed in frame initial
   MySubset will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYSUB.NRLIB/code
   MySingleInteger is now explicitly exposed in frame initial
   MySingleInteger will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYSINT.NRLIB/code
   MyReduce3 is now explicitly exposed in frame initial
   MyReduce3 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYRED3.NRLIB/code
   MyReduce2 is now explicitly exposed in frame initial
   MyReduce2 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYRED2.NRLIB/code
   MyReduce1 is now explicitly exposed in frame initial
   MyReduce1 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYRED1.NRLIB/code
   MyReduce is now explicitly exposed in frame initial
   MyReduce will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYRED.NRLIB/code
   MyPair2 is now explicitly exposed in frame initial
   MyPair2 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYPAIR2.NRLIB/code
   MyPair is now explicitly exposed in frame initial
   MyPair will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYPAIR.NRLIB/code
   MyNonNegativeInteger is now explicitly exposed in frame initial
   MyNonNegativeInteger will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYNNI.NRLIB/code
   MyMonoid1 is now explicitly exposed in frame initial
   MyMonoid1 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYMON1.NRLIB/code
   MyMonoid1& is now explicitly exposed in frame initial
   MyMonoid1& will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYMON1-.NRLIB/code
   MyMonoid is now explicitly exposed in frame initial
   MyMonoid will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYMON.NRLIB/code
   MyMonoid& is now explicitly exposed in frame initial
   MyMonoid& will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYMON-.NRLIB/code
   MyMapping is now explicitly exposed in frame initial
   MyMapping will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYMAP.NRLIB/code
   MyField is now explicitly exposed in frame initial
   MyField will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYFIELD.NRLIB/code
   MyDomain is now explicitly exposed in frame initial
   MyDomain will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MYDOM.NRLIB/code
   MatrixStatPackage is now explicitly exposed in frame initial
   MatrixStatPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MSTAT.NRLIB/code
   Morphism is now explicitly exposed in frame initial
   Morphism will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MORPH.NRLIB/code
   Moo2 is now explicitly exposed in frame initial
   Moo2 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MOO2.NRLIB/code
   Monoid is now explicitly exposed in frame initial
   Monoid will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MONOID.NRLIB/code
   Monadx is now explicitly exposed in frame initial
   Monadx will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MONADX.NRLIB/code
   MkMul is now explicitly exposed in frame initial
   MkMul will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MM.NRLIB/code
   MinimalInitialNotIntegrated is now explicitly exposed in frame
      initial
   MinimalInitialNotIntegrated will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/MINI.NRLIB/code
   MkInt is now explicitly exposed in frame initial
   MkInt will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MI.NRLIB/code
   Maybe2 is now explicitly exposed in frame initial
   Maybe2 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAYBE2.NRLIB/code
   Maybe is now explicitly exposed in frame initial
   Maybe will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAYBE.NRLIB/code
   MappingPackage3x is now explicitly exposed in frame initial
   MappingPackage3x will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAPPK3X.NRLIB/code
   MappingPackage2x is now explicitly exposed in frame initial
   MappingPackage2x will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAPPK2X.NRLIB/code
   MapPackInternalHac3x is now explicitly exposed in frame initial
   MapPackInternalHac3x will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAPHAC3X.NRLIB/code
   MapPackInternalHac2x is now explicitly exposed in frame initial
   MapPackInternalHac2x will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAPHAC2X.NRLIB/code
   Map is now explicitly exposed in frame initial
   Map will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MAP.NRLIB/code
   MkAdd is now explicitly exposed in frame initial
   MkAdd will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/MA.NRLIB/code
   LUDecomposition is now explicitly exposed in frame initial
   LUDecomposition will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/LUD.NRLIB/code
   Inequality is now explicitly exposed in frame initial
   Inequality will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/LT.NRLIB/code
   LittleHttpDeamon is now explicitly exposed in frame initial
   LittleHttpDeamon will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/LHTTPD.NRLIB/code
   LeftFreeModule is now explicitly exposed in frame initial
   LeftFreeModule will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/LFREEMOD.NRLIB/code
   ListAsDomain is now explicitly exposed in frame initial
   ListAsDomain will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/LDOMAIN.NRLIB/code
   JetBundleXExpression is now explicitly exposed in frame initial
   JetBundleXExpression will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/JBX.NRLIB/code
   JetBundleLinearFunction is now explicitly exposed in frame initial
   JetBundleLinearFunction will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/JBLF.NRLIB/code
   JetBundleFunctionCategory is now explicitly exposed in frame initial

   JetBundleFunctionCategory will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/JBFC.NRLIB/code
   JetBundleFunctionCategory& is now explicitly exposed in frame
      initial
   JetBundleFunctionCategory& will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/JBFC-.NRLIB/code
   JetBundleExpression is now explicitly exposed in frame initial
   JetBundleExpression will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/JBE.NRLIB/code
   JetBundleCategory is now explicitly exposed in frame initial
   JetBundleCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/JBC.NRLIB/code
   JetBundleCategory& is now explicitly exposed in frame initial
   JetBundleCategory& will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/JBC-.NRLIB/code
   JetBundle is now explicitly exposed in frame initial
   JetBundle will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/JB.NRLIB/code
   Inverse is now explicitly exposed in frame initial
   Inverse will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INV.NRLIB/code
   InputFormFunctions1 is now explicitly exposed in frame initial
   InputFormFunctions1 will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INFORM1.NRLIB/code
   InputForm is now explicitly exposed in frame initial
   InputForm will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INFORM.NRLIB/code
   Inequality is already explicitly exposed in frame initial
   Inequality will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INEQ.NRLIB/code
   Indeterminants is now explicitly exposed in frame initial
   Indeterminants will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INDETS.NRLIB/code
   Indeterminant is now explicitly exposed in frame initial
   Indeterminant will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/INDET.NRLIB/code
   IndexedJetBundle is now explicitly exposed in frame initial
   IndexedJetBundle will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/IJB.NRLIB/code
   IntegralDomainStatPackage is now explicitly exposed in frame initial

   IntegralDomainStatPackage will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/IDSTAT.NRLIB/code
   Identity is now explicitly exposed in frame initial
   Identity will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/ID.NRLIB/code

   >> System error:
   |Commutative| is invalid as a function.