\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra polset.spad}
\author{Marc Moreno Maza}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{category PSETCAT PolynomialSetCategory}
<<category PSETCAT PolynomialSetCategory>>=
)abbrev category PSETCAT PolynomialSetCategory
++ Author: Marc Moreno Maza
++ Date Created: 04/26/1994
++ Date Last Updated: 12/15/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: 
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References:
++ Description: A category for finite subsets of a polynomial ring.
++ Such a set is only regarded as a set of polynomials and not 
++ identified to the ideal it generates. So two distinct sets may 
++ generate the same the ideal. Furthermore, for \spad{R} being an 
++ integral domain, a set of polynomials may be viewed as a representation
++ of the ideal it generates in the polynomial ring \spad{(R)^(-1) P}, 
++ or the set of its zeros (described for instance by the radical of the 
++ previous ideal, or a split of the associated affine variety) and so on. 
++ So this category provides operations about those different notions.
++ Version: 2 

PolynomialSetCategory(R:Ring, E:OrderedAbelianMonoidSup,_
  VarSet:OrderedSet, P:RecursivePolynomialCategory(R,E,VarSet)): Category == 
    Join(SetCategory,Collection(P),CoercibleTo(List(P))) with
     finiteAggregate
     retractIfCan : List(P) -> Union($,"failed")
         ++ \axiom{retractIfCan(lp)} returns an element of the domain whose elements
         ++ are the members of \axiom{lp} if such an element exists, otherwise
         ++ \axiom{"failed"} is returned.
     retract : List(P) -> $
         ++ \axiom{retract(lp)} returns an element of the domain whose elements
         ++ are the members of \axiom{lp} if such an element exists, otherwise
         ++ an error is produced.
     mvar : $ -> VarSet
         ++  \axiom{mvar(ps)} returns the main variable of the non constant polynomial
         ++  with the greatest main variable, if any, else an error is returned.
     variables : $ -> List VarSet
         ++  \axiom{variables(ps)} returns the decreasingly sorted list of the
         ++  variables which are variables of some polynomial in \axiom{ps}.
     mainVariables : $  -> List VarSet
         ++ \axiom{mainVariables(ps)} returns the decreasingly sorted list of the
         ++ variables which are main variables of some polynomial in \axiom{ps}.
     mainVariable? : (VarSet,$) -> Boolean
         ++ \axiom{mainVariable?(v,ps)} returns true iff \axiom{v} is the main variable 
         ++ of some polynomial in \axiom{ps}.
     collectUnder : ($,VarSet) -> $
         ++ \axiom{collectUnder(ps,v)} returns the set consisting of the 
         ++ polynomials of \axiom{ps} with main variable less than \axiom{v}.
     collect : ($,VarSet) -> $
         ++ \axiom{collect(ps,v)}  returns the set consisting of the 
         ++ polynomials of \axiom{ps} with \axiom{v} as main variable.
     collectUpper : ($,VarSet) -> $
         ++ \axiom{collectUpper(ps,v)} returns the set consisting of the 
         ++ polynomials of \axiom{ps} with main variable greater than \axiom{v}.
     sort : ($,VarSet) -> Record(under:$,floor:$,upper:$)
         ++ \axiom{sort(v,ps)} returns \axiom{us,vs,ws} such that \axiom{us}
         ++ is \axiom{collectUnder(ps,v)}, \axiom{vs} is \axiom{collect(ps,v)}
         ++ and \axiom{ws} is \axiom{collectUpper(ps,v)}. 
     trivialIdeal?: $ -> Boolean
         ++ \axiom{trivialIdeal?(ps)} returns true iff \axiom{ps} does
         ++ not contain non-zero elements.
     if R has IntegralDomain
     then
       roughBase? : $ -> Boolean
           ++ \axiom{roughBase?(ps)} returns true iff for every pair \axiom{{p,q}}
           ++ of polynomials in \axiom{ps} their leading monomials are 
           ++ relatively prime.
       roughSubIdeal?  : ($,$) -> Boolean
           ++  \axiom{roughSubIdeal?(ps1,ps2)} returns true iff it can proved 
           ++ that all polynomials in  \axiom{ps1} lie in the ideal generated by 
           ++ \axiom{ps2} in  \axiom{\axiom{(R)^(-1) P}} without computing Groebner bases.
       roughEqualIdeals? : ($,$) -> Boolean
           ++ \axiom{roughEqualIdeals?(ps1,ps2)} returns true iff it can
           ++ proved that \axiom{ps1} and \axiom{ps2} generate the same ideal
           ++ in \axiom{(R)^(-1) P} without computing Groebner bases.
       roughUnitIdeal? : $ -> Boolean
           ++ \axiom{roughUnitIdeal?(ps)} returns true iff \axiom{ps} contains some
           ++ non null element lying in the base ring \axiom{R}.
       headRemainder : (P,$) -> Record(num:P,den:R)
           ++ \axiom{headRemainder(a,ps)} returns \axiom{[b,r]} such that the leading
           ++ monomial of \axiom{b} is reduced in the sense of Groebner bases w.r.t.
           ++ \axiom{ps} and \axiom{r*a - b} lies in the ideal generated by \axiom{ps}.
       remainder : (P,$) -> Record(rnum:R,polnum:P,den:R)
           ++ \axiom{remainder(a,ps)} returns \axiom{[c,b,r]} such that \axiom{b} is fully 
           ++ reduced in the sense of Groebner bases w.r.t. \axiom{ps}, 
           ++ \axiom{r*a - c*b} lies in the ideal generated by \axiom{ps}.
           ++ Furthermore, if \axiom{R} is a gcd-domain, \axiom{b} is primitive.
       rewriteIdealWithHeadRemainder : (List(P),$) -> List(P)
           ++ \axiom{rewriteIdealWithHeadRemainder(lp,cs)} returns \axiom{lr} such that
           ++ the leading monomial of every polynomial in \axiom{lr} is reduced
           ++ in the sense of Groebner bases w.r.t. \axiom{cs} and \axiom{(lp,cs)}
           ++ and \axiom{(lr,cs)} generate the same ideal in \axiom{(R)^(-1) P}.
       rewriteIdealWithRemainder : (List(P),$) -> List(P)
           ++ \axiom{rewriteIdealWithRemainder(lp,cs)} returns \axiom{lr} such that
           ++ every polynomial in \axiom{lr} is fully reduced in the sense 
           ++ of Groebner bases w.r.t. \axiom{cs} and \axiom{(lp,cs)} and 
           ++ \axiom{(lr,cs)} generate the same ideal in \axiom{(R)^(-1) P}.
       triangular? : $ -> Boolean
           ++ \axiom{triangular?(ps)} returns true iff \axiom{ps} is a triangular set,
           ++ i.e. two distinct polynomials have distinct main variables
           ++ and no constant lies in \axiom{ps}.

  add

     NNI ==> NonNegativeInteger
     B ==> Boolean

     elements: $ -> List(P)

     elements(ps:$):List(P) ==
       lp : List(P) := members(ps)$$

     variables1(lp:List(P)):(List VarSet) ==
       lvars : List(List(VarSet)) := [variables(p)$P for p in lp]
       sort(#1 > #2, removeDuplicates(concat(lvars)$List(VarSet)))

     variables2(lp:List(P)):(List VarSet) ==
       lvars : List(VarSet) := [mvar(p)$P for p in lp]
       sort(#1 > #2, removeDuplicates(lvars)$List(VarSet))

     variables (ps:$) ==
       variables1(elements(ps))

     mainVariables (ps:$) ==
       variables2(remove(ground?,elements(ps)))

     mainVariable? (v,ps) ==
       lp : List(P) := remove(ground?,elements(ps))
       while (not empty? lp) and (not (mvar(first(lp)) = v)) repeat
         lp := rest lp
       (not empty? lp)

     collectUnder (ps,v) ==
       lp : List P := elements(ps)
       lq : List P := []
       while (not empty? lp) repeat
         p := first lp
         lp := rest lp
         if (ground?(p)) or (mvar(p) < v)
           then
             lq := cons(p,lq)
       construct(lq)$$

     collectUpper (ps,v) ==
       lp : List P := elements(ps)
       lq : List P := []
       while (not empty? lp) repeat
         p := first lp
         lp := rest lp
         if (not ground?(p)) and (mvar(p) > v)
           then
             lq := cons(p,lq)
       construct(lq)$$

     collect (ps,v) ==
       lp : List P := elements(ps)
       lq : List P := []
       while (not empty? lp) repeat
         p := first lp
         lp := rest lp
         if (not ground?(p)) and (mvar(p) = v)
           then
             lq := cons(p,lq)
       construct(lq)$$

     sort (ps,v) ==
       lp : List P := elements(ps)
       us : List P := []
       vs : List P := []
       ws : List P := []
       while (not empty? lp) repeat
         p := first lp
         lp := rest lp
         if (ground?(p)) or (mvar(p) < v)
           then
             us := cons(p,us)
           else
             if (mvar(p) = v)
               then
                 vs := cons(p,vs)
               else
                 ws := cons(p,ws)
       [construct(us)$$,construct(vs)$$,construct(ws)$$]$Record(under:$,floor:$,upper:$)

     ps1 = ps2 ==
       {p for p in elements(ps1)} =$(Set P) {p for p in elements(ps2)}

     exactQuo : (R,R) -> R

     localInf? (p:P,q:P):B ==
       degree(p) <$E degree(q)

     localTriangular? (lp:List(P)):B ==
       lp := remove(zero?, lp)
       empty? lp => true
       any? (ground?, lp) => false
       lp := sort(mvar(#1)$P > mvar(#2)$P, lp)
       p,q : P
       p := first lp
       lp := rest lp
       while (not empty? lp) and (mvar(p) > mvar((q := first(lp)))) repeat
         p := q
         lp := rest lp
       empty? lp

     triangular? ps ==
       localTriangular? elements ps

     trivialIdeal? ps ==
       empty?(remove(zero?,elements(ps))$(List(P)))$(List(P))

     if R has IntegralDomain
     then

       roughUnitIdeal? ps ==
         any?(ground?,remove(zero?,elements(ps))$(List(P)))$(List P)

       relativelyPrimeLeadingMonomials? (p:P,q:P):B ==
         dp : E := degree(p)
         dq : E := degree(q)
         (sup(dp,dq)$E =$E dp +$E dq)@B

       roughBase? ps ==
         lp := remove(zero?,elements(ps))$(List(P))
         empty? lp => true
         rB? : B := true
         while (not empty? lp) and rB? repeat
           p := first lp
           lp := rest lp
           copylp := lp
           while (not empty? copylp) and rB? repeat
             rB? := relativelyPrimeLeadingMonomials?(p,first(copylp))
             copylp := rest copylp
         rB?

       roughSubIdeal?(ps1,ps2) ==
         lp: List(P) := rewriteIdealWithRemainder(elements(ps1),ps2)
         empty? (remove(zero?,lp))

       roughEqualIdeals? (ps1,ps2) ==
         ps1 =$$ ps2 => true
         roughSubIdeal?(ps1,ps2) and roughSubIdeal?(ps2,ps1)

     if (R has GcdDomain) and (VarSet has ConvertibleTo (Symbol))
     then

       LPR ==> List Polynomial R
       LS ==> List Symbol

       if R has EuclideanDomain
         then
           exactQuo(r:R,s:R):R ==
             r quo$R s
         else
           exactQuo(r:R,s:R):R ==
             (r exquo$R s)::R

       headRemainder (a,ps) ==
         lp1 : List(P) := remove(zero?, elements(ps))$(List(P))
         empty? lp1 => [a,1$R]
         any?(ground?,lp1) => [reductum(a),1$R]
         r : R := 1$R
         lp1 := sort(localInf?, reverse elements(ps))
         lp2 := lp1
         e : Union(E, "failed")
         while (not zero? a) and (not empty? lp2) repeat
           p := first lp2
           if ((e:= subtractIfCan(degree(a),degree(p))) case E)
             then
               g := gcd((lca := leadingCoefficient(a)),(lcp := leadingCoefficient(p)))$R
               (lca,lcp) := (exactQuo(lca,g),exactQuo(lcp,g))
               a := lcp * reductum(a) - monomial(lca, e::E)$P * reductum(p)
               r := r * lcp
               lp2 := lp1
             else
               lp2 := rest lp2
         [a,r]

       makeIrreducible! (frac:Record(num:P,den:R)):Record(num:P,den:R) ==
         g := gcd(frac.den,frac.num)$P
--         one? g => frac
         (g = 1) => frac
         frac.num := exactQuotient!(frac.num,g)
         frac.den := exactQuo(frac.den,g)
         frac

       remainder (a,ps) ==
         hRa := makeIrreducible! headRemainder (a,ps)
         a := hRa.num
         r : R := hRa.den
         zero? a => [1$R,a,r]
         b : P := monomial(1$R,degree(a))$P
         c : R := leadingCoefficient(a)
         while not zero?(a := reductum a) repeat
           hRa := makeIrreducible!  headRemainder (a,ps)
           a := hRa.num
           r := r * hRa.den
           g := gcd(c,(lca := leadingCoefficient(a)))$R
           b := ((hRa.den) * exactQuo(c,g)) * b + monomial(exactQuo(lca,g),degree(a))$P
           c := g
         [c,b,r]

       rewriteIdealWithHeadRemainder(ps,cs) ==
         trivialIdeal? cs => ps
         roughUnitIdeal? cs => [0$P]
         ps := remove(zero?,ps)
         empty? ps => ps
         any?(ground?,ps) => [1$P]
         rs : List P := []
         while not empty? ps repeat
           p := first ps
           ps := rest ps
           p := (headRemainder(p,cs)).num
           if not zero? p
             then 
               if ground? p
                 then
                   ps := []
                   rs := [1$P]
                 else
                   primitivePart! p
                   rs := cons(p,rs)
         removeDuplicates rs

       rewriteIdealWithRemainder(ps,cs) ==
         trivialIdeal? cs => ps
         roughUnitIdeal? cs => [0$P]
         ps := remove(zero?,ps)
         empty? ps => ps
         any?(ground?,ps) => [1$P]
         rs : List P := []
         while not empty? ps repeat
           p := first ps
           ps := rest ps
           p := (remainder(p,cs)).polnum
           if not zero? p
             then 
               if ground? p
                 then
                   ps := []
                   rs := [1$P]
                 else
                   rs := cons(unitCanonical(p),rs)
         removeDuplicates rs

@
\section{PSETCAT.lsp BOOTSTRAP}
{\bf PSETCAT} depends on a chain of files. We need to break this cycle to build
the algebra. So we keep a cached copy of the translated {\bf PSETCAT}
category which we can write into the {\bf MID} directory. We compile 
the lisp code and copy the {\bf PSETCAT.o} file to the {\bf OUT} directory.
This is eventually forcibly replaced by a recompiled version. 

Note that this code is not included in the generated catdef.spad file.

<<PSETCAT.lsp BOOTSTRAP>>=

(|/VERSIONCHECK| 2) 

(SETQ |PolynomialSetCategory;CAT| (QUOTE NIL)) 

(SETQ |PolynomialSetCategory;AL| (QUOTE NIL)) 

(DEFUN |PolynomialSetCategory| (|&REST| #1=#:G82375 |&AUX| #2=#:G82373) (DSETQ #2# #1#) (LET (#3=#:G82374) (COND ((SETQ #3# (|assoc| (|devaluateList| #2#) |PolynomialSetCategory;AL|)) (CDR #3#)) (T (SETQ |PolynomialSetCategory;AL| (|cons5| (CONS (|devaluateList| #2#) (SETQ #3# (APPLY (FUNCTION |PolynomialSetCategory;|) #2#))) |PolynomialSetCategory;AL|)) #3#)))) 

(DEFUN |PolynomialSetCategory;| (|t#1| |t#2| |t#3| |t#4|) (PROG (#1=#:G82372) (RETURN (PROG1 (LETT #1# (|sublisV| (PAIR (QUOTE (|t#1| |t#2| |t#3| |t#4|)) (LIST (|devaluate| |t#1|) (|devaluate| |t#2|) (|devaluate| |t#3|) (|devaluate| |t#4|))) (|sublisV| (PAIR (QUOTE (#2=#:G82371)) (LIST (QUOTE (|List| |t#4|)))) (COND (|PolynomialSetCategory;CAT|) ((QUOTE T) (LETT |PolynomialSetCategory;CAT| (|Join| (|SetCategory|) (|Collection| (QUOTE |t#4|)) (|CoercibleTo| (QUOTE #2#)) (|mkCategory| (QUOTE |domain|) (QUOTE (((|retractIfCan| ((|Union| |$| "failed") (|List| |t#4|))) T) ((|retract| (|$| (|List| |t#4|))) T) ((|mvar| (|t#3| |$|)) T) ((|variables| ((|List| |t#3|) |$|)) T) ((|mainVariables| ((|List| |t#3|) |$|)) T) ((|mainVariable?| ((|Boolean|) |t#3| |$|)) T) ((|collectUnder| (|$| |$| |t#3|)) T) ((|collect| (|$| |$| |t#3|)) T) ((|collectUpper| (|$| |$| |t#3|)) T) ((|sort| ((|Record| (|:| |under| |$|) (|:| |floor| |$|) (|:| |upper| |$|)) |$| |t#3|)) T) ((|trivialIdeal?| ((|Boolean|) |$|)) T) ((|roughBase?| ((|Boolean|) |$|)) (|has| |t#1| (|IntegralDomain|))) ((|roughSubIdeal?| ((|Boolean|) |$| |$|)) (|has| |t#1| (|IntegralDomain|))) ((|roughEqualIdeals?| ((|Boolean|) |$| |$|)) (|has| |t#1| (|IntegralDomain|))) ((|roughUnitIdeal?| ((|Boolean|) |$|)) (|has| |t#1| (|IntegralDomain|))) ((|headRemainder| ((|Record| (|:| |num| |t#4|) (|:| |den| |t#1|)) |t#4| |$|)) (|has| |t#1| (|IntegralDomain|))) ((|remainder| ((|Record| (|:| |rnum| |t#1|) (|:| |polnum| |t#4|) (|:| |den| |t#1|)) |t#4| |$|)) (|has| |t#1| (|IntegralDomain|))) ((|rewriteIdealWithHeadRemainder| ((|List| |t#4|) (|List| |t#4|) |$|)) (|has| |t#1| (|IntegralDomain|))) ((|rewriteIdealWithRemainder| ((|List| |t#4|) (|List| |t#4|) |$|)) (|has| |t#1| (|IntegralDomain|))) ((|triangular?| ((|Boolean|) |$|)) (|has| |t#1| (|IntegralDomain|))))) (QUOTE ((|finiteAggregate| T))) (QUOTE ((|Boolean|) (|List| |t#4|) (|List| |t#3|))) NIL)) . #3=(|PolynomialSetCategory|)))))) . #3#) (SETELT #1# 0 (LIST (QUOTE |PolynomialSetCategory|) (|devaluate| |t#1|) (|devaluate| |t#2|) (|devaluate| |t#3|) (|devaluate| |t#4|))))))) 
@
\section{PSETCAT-.lsp BOOTSTRAP}
{\bf PSETCAT-} depends on {\bf PSETCAT}. We need to break this cycle to build
the algebra. So we keep a cached copy of the translated {\bf PSETCAT-}
category which we can write into the {\bf MID} directory. We compile 
the lisp code and copy the {\bf PSETCAT-.o} file to the {\bf OUT} directory.
This is eventually forcibly replaced by a recompiled version. 

Note that this code is not included in the generated catdef.spad file.

<<PSETCAT-.lsp BOOTSTRAP>>=

(|/VERSIONCHECK| 2) 

(DEFUN |PSETCAT-;elements| (|ps| |$|) (PROG (|lp|) (RETURN (LETT |lp| (SPADCALL |ps| (QREFELT |$| 12)) |PSETCAT-;elements|)))) 

(DEFUN |PSETCAT-;variables1| (|lp| |$|) (PROG (#1=#:G82392 |p| #2=#:G82393 |lvars|) (RETURN (SEQ (LETT |lvars| (PROGN (LETT #1# NIL |PSETCAT-;variables1|) (SEQ (LETT |p| NIL |PSETCAT-;variables1|) (LETT #2# |lp| |PSETCAT-;variables1|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |p| (CAR #2#) |PSETCAT-;variables1|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |p| (QREFELT |$| 14)) #1#) |PSETCAT-;variables1|))) (LETT #2# (CDR #2#) |PSETCAT-;variables1|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) |PSETCAT-;variables1|) (EXIT (SPADCALL (CONS (FUNCTION |PSETCAT-;variables1!0|) |$|) (SPADCALL (SPADCALL |lvars| (QREFELT |$| 18)) (QREFELT |$| 19)) (QREFELT |$| 21))))))) 

(DEFUN |PSETCAT-;variables1!0| (|#1| |#2| |$|) (SPADCALL |#2| |#1| (QREFELT |$| 16))) 

(DEFUN |PSETCAT-;variables2| (|lp| |$|) (PROG (#1=#:G82397 |p| #2=#:G82398 |lvars|) (RETURN (SEQ (LETT |lvars| (PROGN (LETT #1# NIL |PSETCAT-;variables2|) (SEQ (LETT |p| NIL |PSETCAT-;variables2|) (LETT #2# |lp| |PSETCAT-;variables2|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |p| (CAR #2#) |PSETCAT-;variables2|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS (SPADCALL |p| (QREFELT |$| 22)) #1#) |PSETCAT-;variables2|))) (LETT #2# (CDR #2#) |PSETCAT-;variables2|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) |PSETCAT-;variables2|) (EXIT (SPADCALL (CONS (FUNCTION |PSETCAT-;variables2!0|) |$|) (SPADCALL |lvars| (QREFELT |$| 19)) (QREFELT |$| 21))))))) 

(DEFUN |PSETCAT-;variables2!0| (|#1| |#2| |$|) (SPADCALL |#2| |#1| (QREFELT |$| 16))) 

(DEFUN |PSETCAT-;variables;SL;4| (|ps| |$|) (|PSETCAT-;variables1| (|PSETCAT-;elements| |ps| |$|) |$|)) 

(DEFUN |PSETCAT-;mainVariables;SL;5| (|ps| |$|) (|PSETCAT-;variables2| (SPADCALL (ELT |$| 24) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)) |$|)) 

(DEFUN |PSETCAT-;mainVariable?;VarSetSB;6| (|v| |ps| |$|) (PROG (|lp|) (RETURN (SEQ (LETT |lp| (SPADCALL (ELT |$| 24) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)) |PSETCAT-;mainVariable?;VarSetSB;6|) (SEQ G190 (COND ((NULL (COND ((OR (NULL |lp|) (SPADCALL (SPADCALL (|SPADfirst| |lp|) (QREFELT |$| 22)) |v| (QREFELT |$| 28))) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (EXIT (LETT |lp| (CDR |lp|) |PSETCAT-;mainVariable?;VarSetSB;6|))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))))))) 

(DEFUN |PSETCAT-;collectUnder;SVarSetS;7| (|ps| |v| |$|) (PROG (|p| |lp| |lq|) (RETURN (SEQ (LETT |lp| (|PSETCAT-;elements| |ps| |$|) |PSETCAT-;collectUnder;SVarSetS;7|) (LETT |lq| NIL |PSETCAT-;collectUnder;SVarSetS;7|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;collectUnder;SVarSetS;7|) (LETT |lp| (CDR |lp|) |PSETCAT-;collectUnder;SVarSetS;7|) (EXIT (COND ((OR (SPADCALL |p| (QREFELT |$| 24)) (SPADCALL (SPADCALL |p| (QREFELT |$| 22)) |v| (QREFELT |$| 16))) (LETT |lq| (CONS |p| |lq|) |PSETCAT-;collectUnder;SVarSetS;7|))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (SPADCALL |lq| (QREFELT |$| 30))))))) 

(DEFUN |PSETCAT-;collectUpper;SVarSetS;8| (|ps| |v| |$|) (PROG (|p| |lp| |lq|) (RETURN (SEQ (LETT |lp| (|PSETCAT-;elements| |ps| |$|) |PSETCAT-;collectUpper;SVarSetS;8|) (LETT |lq| NIL |PSETCAT-;collectUpper;SVarSetS;8|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;collectUpper;SVarSetS;8|) (LETT |lp| (CDR |lp|) |PSETCAT-;collectUpper;SVarSetS;8|) (EXIT (COND ((NULL (SPADCALL |p| (QREFELT |$| 24))) (COND ((SPADCALL |v| (SPADCALL |p| (QREFELT |$| 22)) (QREFELT |$| 16)) (LETT |lq| (CONS |p| |lq|) |PSETCAT-;collectUpper;SVarSetS;8|))))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (SPADCALL |lq| (QREFELT |$| 30))))))) 

(DEFUN |PSETCAT-;collect;SVarSetS;9| (|ps| |v| |$|) (PROG (|p| |lp| |lq|) (RETURN (SEQ (LETT |lp| (|PSETCAT-;elements| |ps| |$|) |PSETCAT-;collect;SVarSetS;9|) (LETT |lq| NIL |PSETCAT-;collect;SVarSetS;9|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;collect;SVarSetS;9|) (LETT |lp| (CDR |lp|) |PSETCAT-;collect;SVarSetS;9|) (EXIT (COND ((NULL (SPADCALL |p| (QREFELT |$| 24))) (COND ((SPADCALL (SPADCALL |p| (QREFELT |$| 22)) |v| (QREFELT |$| 28)) (LETT |lq| (CONS |p| |lq|) |PSETCAT-;collect;SVarSetS;9|))))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (SPADCALL |lq| (QREFELT |$| 30))))))) 

(DEFUN |PSETCAT-;sort;SVarSetR;10| (|ps| |v| |$|) (PROG (|p| |lp| |us| |vs| |ws|) (RETURN (SEQ (LETT |lp| (|PSETCAT-;elements| |ps| |$|) |PSETCAT-;sort;SVarSetR;10|) (LETT |us| NIL |PSETCAT-;sort;SVarSetR;10|) (LETT |vs| NIL |PSETCAT-;sort;SVarSetR;10|) (LETT |ws| NIL |PSETCAT-;sort;SVarSetR;10|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;sort;SVarSetR;10|) (LETT |lp| (CDR |lp|) |PSETCAT-;sort;SVarSetR;10|) (EXIT (COND ((OR (SPADCALL |p| (QREFELT |$| 24)) (SPADCALL (SPADCALL |p| (QREFELT |$| 22)) |v| (QREFELT |$| 16))) (LETT |us| (CONS |p| |us|) |PSETCAT-;sort;SVarSetR;10|)) ((SPADCALL (SPADCALL |p| (QREFELT |$| 22)) |v| (QREFELT |$| 28)) (LETT |vs| (CONS |p| |vs|) |PSETCAT-;sort;SVarSetR;10|)) ((QUOTE T) (LETT |ws| (CONS |p| |ws|) |PSETCAT-;sort;SVarSetR;10|))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (VECTOR (SPADCALL |us| (QREFELT |$| 30)) (SPADCALL |vs| (QREFELT |$| 30)) (SPADCALL |ws| (QREFELT |$| 30)))))))) 

(DEFUN |PSETCAT-;=;2SB;11| (|ps1| |ps2| |$|) (PROG (#1=#:G82439 #2=#:G82440 #3=#:G82437 |p| #4=#:G82438) (RETURN (SEQ (SPADCALL (SPADCALL (PROGN (LETT #1# NIL |PSETCAT-;=;2SB;11|) (SEQ (LETT |p| NIL |PSETCAT-;=;2SB;11|) (LETT #2# (|PSETCAT-;elements| |ps1| |$|) |PSETCAT-;=;2SB;11|) G190 (COND ((OR (ATOM #2#) (PROGN (LETT |p| (CAR #2#) |PSETCAT-;=;2SB;11|) NIL)) (GO G191))) (SEQ (EXIT (LETT #1# (CONS |p| #1#) |PSETCAT-;=;2SB;11|))) (LETT #2# (CDR #2#) |PSETCAT-;=;2SB;11|) (GO G190) G191 (EXIT (NREVERSE0 #1#)))) (QREFELT |$| 37)) (SPADCALL (PROGN (LETT #3# NIL |PSETCAT-;=;2SB;11|) (SEQ (LETT |p| NIL |PSETCAT-;=;2SB;11|) (LETT #4# (|PSETCAT-;elements| |ps2| |$|) |PSETCAT-;=;2SB;11|) G190 (COND ((OR (ATOM #4#) (PROGN (LETT |p| (CAR #4#) |PSETCAT-;=;2SB;11|) NIL)) (GO G191))) (SEQ (EXIT (LETT #3# (CONS |p| #3#) |PSETCAT-;=;2SB;11|))) (LETT #4# (CDR #4#) |PSETCAT-;=;2SB;11|) (GO G190) G191 (EXIT (NREVERSE0 #3#)))) (QREFELT |$| 37)) (QREFELT |$| 38)))))) 

(DEFUN |PSETCAT-;localInf?| (|p| |q| |$|) (SPADCALL (SPADCALL |p| (QREFELT |$| 40)) (SPADCALL |q| (QREFELT |$| 40)) (QREFELT |$| 41))) 

(DEFUN |PSETCAT-;localTriangular?| (|lp| |$|) (PROG (|q| |p|) (RETURN (SEQ (LETT |lp| (SPADCALL (ELT |$| 42) |lp| (QREFELT |$| 26)) |PSETCAT-;localTriangular?|) (EXIT (COND ((NULL |lp|) (QUOTE T)) ((SPADCALL (ELT |$| 24) |lp| (QREFELT |$| 43)) (QUOTE NIL)) ((QUOTE T) (SEQ (LETT |lp| (SPADCALL (CONS (FUNCTION |PSETCAT-;localTriangular?!0|) |$|) |lp| (QREFELT |$| 45)) |PSETCAT-;localTriangular?|) (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;localTriangular?|) (LETT |lp| (CDR |lp|) |PSETCAT-;localTriangular?|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) (SPADCALL (SPADCALL (LETT |q| (|SPADfirst| |lp|) |PSETCAT-;localTriangular?|) (QREFELT |$| 22)) (SPADCALL |p| (QREFELT |$| 22)) (QREFELT |$| 16))))) (GO G191))) (SEQ (LETT |p| |q| |PSETCAT-;localTriangular?|) (EXIT (LETT |lp| (CDR |lp|) |PSETCAT-;localTriangular?|))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (NULL |lp|)))))))))) 

(DEFUN |PSETCAT-;localTriangular?!0| (|#1| |#2| |$|) (SPADCALL (SPADCALL |#2| (QREFELT |$| 22)) (SPADCALL |#1| (QREFELT |$| 22)) (QREFELT |$| 16))) 

(DEFUN |PSETCAT-;triangular?;SB;14| (|ps| |$|) (|PSETCAT-;localTriangular?| (|PSETCAT-;elements| |ps| |$|) |$|)) 

(DEFUN |PSETCAT-;trivialIdeal?;SB;15| (|ps| |$|) (NULL (SPADCALL (ELT |$| 42) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)))) 

(DEFUN |PSETCAT-;roughUnitIdeal?;SB;16| (|ps| |$|) (SPADCALL (ELT |$| 24) (SPADCALL (ELT |$| 42) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)) (QREFELT |$| 43))) 

(DEFUN |PSETCAT-;relativelyPrimeLeadingMonomials?| (|p| |q| |$|) (PROG (|dp| |dq|) (RETURN (SEQ (LETT |dp| (SPADCALL |p| (QREFELT |$| 40)) |PSETCAT-;relativelyPrimeLeadingMonomials?|) (LETT |dq| (SPADCALL |q| (QREFELT |$| 40)) |PSETCAT-;relativelyPrimeLeadingMonomials?|) (EXIT (SPADCALL (SPADCALL |dp| |dq| (QREFELT |$| 49)) (SPADCALL |dp| |dq| (QREFELT |$| 50)) (QREFELT |$| 51))))))) 

(DEFUN |PSETCAT-;roughBase?;SB;18| (|ps| |$|) (PROG (|p| |lp| |rB?| |copylp|) (RETURN (SEQ (LETT |lp| (SPADCALL (ELT |$| 42) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)) |PSETCAT-;roughBase?;SB;18|) (EXIT (COND ((NULL |lp|) (QUOTE T)) ((QUOTE T) (SEQ (LETT |rB?| (QUOTE T) |PSETCAT-;roughBase?;SB;18|) (SEQ G190 (COND ((NULL (COND ((NULL |lp|) (QUOTE NIL)) ((QUOTE T) |rB?|))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp|) |PSETCAT-;roughBase?;SB;18|) (LETT |lp| (CDR |lp|) |PSETCAT-;roughBase?;SB;18|) (LETT |copylp| |lp| |PSETCAT-;roughBase?;SB;18|) (EXIT (SEQ G190 (COND ((NULL (COND ((NULL |copylp|) (QUOTE NIL)) ((QUOTE T) |rB?|))) (GO G191))) (SEQ (LETT |rB?| (|PSETCAT-;relativelyPrimeLeadingMonomials?| |p| (|SPADfirst| |copylp|) |$|) |PSETCAT-;roughBase?;SB;18|) (EXIT (LETT |copylp| (CDR |copylp|) |PSETCAT-;roughBase?;SB;18|))) NIL (GO G190) G191 (EXIT NIL)))) NIL (GO G190) G191 (EXIT NIL)) (EXIT |rB?|))))))))) 

(DEFUN |PSETCAT-;roughSubIdeal?;2SB;19| (|ps1| |ps2| |$|) (PROG (|lp|) (RETURN (SEQ (LETT |lp| (SPADCALL (|PSETCAT-;elements| |ps1| |$|) |ps2| (QREFELT |$| 53)) |PSETCAT-;roughSubIdeal?;2SB;19|) (EXIT (NULL (SPADCALL (ELT |$| 42) |lp| (QREFELT |$| 26)))))))) 

(DEFUN |PSETCAT-;roughEqualIdeals?;2SB;20| (|ps1| |ps2| |$|) (COND ((SPADCALL |ps1| |ps2| (QREFELT |$| 55)) (QUOTE T)) ((SPADCALL |ps1| |ps2| (QREFELT |$| 56)) (SPADCALL |ps2| |ps1| (QREFELT |$| 56))) ((QUOTE T) (QUOTE NIL)))) 

(DEFUN |PSETCAT-;exactQuo| (|r| |s| |$|) (SPADCALL |r| |s| (QREFELT |$| 58))) 

(DEFUN |PSETCAT-;exactQuo| (|r| |s| |$|) (PROG (#1=#:G82473) (RETURN (PROG2 (LETT #1# (SPADCALL |r| |s| (QREFELT |$| 60)) |PSETCAT-;exactQuo|) (QCDR #1#) (|check-union| (QEQCAR #1# 0) (QREFELT |$| 7) #1#))))) 

(DEFUN |PSETCAT-;headRemainder;PSR;23| (|a| |ps| |$|) (PROG (|lp1| |p| |e| |g| |#G47| |#G48| |lca| |lcp| |r| |lp2|) (RETURN (SEQ (LETT |lp1| (SPADCALL (ELT |$| 42) (|PSETCAT-;elements| |ps| |$|) (QREFELT |$| 26)) |PSETCAT-;headRemainder;PSR;23|) (EXIT (COND ((NULL |lp1|) (CONS |a| (|spadConstant| |$| 61))) ((SPADCALL (ELT |$| 24) |lp1| (QREFELT |$| 43)) (CONS (SPADCALL |a| (QREFELT |$| 62)) (|spadConstant| |$| 61))) ((QUOTE T) (SEQ (LETT |r| (|spadConstant| |$| 61) |PSETCAT-;headRemainder;PSR;23|) (LETT |lp1| (SPADCALL (CONS (|function| |PSETCAT-;localInf?|) |$|) (REVERSE (|PSETCAT-;elements| |ps| |$|)) (QREFELT |$| 45)) |PSETCAT-;headRemainder;PSR;23|) (LETT |lp2| |lp1| |PSETCAT-;headRemainder;PSR;23|) (SEQ G190 (COND ((NULL (COND ((OR (SPADCALL |a| (QREFELT |$| 42)) (NULL |lp2|)) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |lp2|) |PSETCAT-;headRemainder;PSR;23|) (LETT |e| (SPADCALL (SPADCALL |a| (QREFELT |$| 40)) (SPADCALL |p| (QREFELT |$| 40)) (QREFELT |$| 63)) |PSETCAT-;headRemainder;PSR;23|) (EXIT (COND ((QEQCAR |e| 0) (SEQ (LETT |g| (SPADCALL (LETT |lca| (SPADCALL |a| (QREFELT |$| 64)) |PSETCAT-;headRemainder;PSR;23|) (LETT |lcp| (SPADCALL |p| (QREFELT |$| 64)) |PSETCAT-;headRemainder;PSR;23|) (QREFELT |$| 65)) |PSETCAT-;headRemainder;PSR;23|) (PROGN (LETT |#G47| (|PSETCAT-;exactQuo| |lca| |g| |$|) |PSETCAT-;headRemainder;PSR;23|) (LETT |#G48| (|PSETCAT-;exactQuo| |lcp| |g| |$|) |PSETCAT-;headRemainder;PSR;23|) (LETT |lca| |#G47| |PSETCAT-;headRemainder;PSR;23|) (LETT |lcp| |#G48| |PSETCAT-;headRemainder;PSR;23|)) (LETT |a| (SPADCALL (SPADCALL |lcp| (SPADCALL |a| (QREFELT |$| 62)) (QREFELT |$| 66)) (SPADCALL (SPADCALL |lca| (QCDR |e|) (QREFELT |$| 67)) (SPADCALL |p| (QREFELT |$| 62)) (QREFELT |$| 68)) (QREFELT |$| 69)) |PSETCAT-;headRemainder;PSR;23|) (LETT |r| (SPADCALL |r| |lcp| (QREFELT |$| 70)) |PSETCAT-;headRemainder;PSR;23|) (EXIT (LETT |lp2| |lp1| |PSETCAT-;headRemainder;PSR;23|)))) ((QUOTE T) (LETT |lp2| (CDR |lp2|) |PSETCAT-;headRemainder;PSR;23|))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (CONS |a| |r|)))))))))) 

(DEFUN |PSETCAT-;makeIrreducible!| (|frac| |$|) (PROG (|g|) (RETURN (SEQ (LETT |g| (SPADCALL (QCDR |frac|) (QCAR |frac|) (QREFELT |$| 73)) |PSETCAT-;makeIrreducible!|) (EXIT (COND ((SPADCALL |g| (QREFELT |$| 74)) |frac|) ((QUOTE T) (SEQ (PROGN (RPLACA |frac| (SPADCALL (QCAR |frac|) |g| (QREFELT |$| 75))) (QCAR |frac|)) (PROGN (RPLACD |frac| (|PSETCAT-;exactQuo| (QCDR |frac|) |g| |$|)) (QCDR |frac|)) (EXIT |frac|))))))))) 

(DEFUN |PSETCAT-;remainder;PSR;25| (|a| |ps| |$|) (PROG (|hRa| |r| |lca| |g| |b| |c|) (RETURN (SEQ (LETT |hRa| (|PSETCAT-;makeIrreducible!| (SPADCALL |a| |ps| (QREFELT |$| 76)) |$|) |PSETCAT-;remainder;PSR;25|) (LETT |a| (QCAR |hRa|) |PSETCAT-;remainder;PSR;25|) (LETT |r| (QCDR |hRa|) |PSETCAT-;remainder;PSR;25|) (EXIT (COND ((SPADCALL |a| (QREFELT |$| 42)) (VECTOR (|spadConstant| |$| 61) |a| |r|)) ((QUOTE T) (SEQ (LETT |b| (SPADCALL (|spadConstant| |$| 61) (SPADCALL |a| (QREFELT |$| 40)) (QREFELT |$| 67)) |PSETCAT-;remainder;PSR;25|) (LETT |c| (SPADCALL |a| (QREFELT |$| 64)) |PSETCAT-;remainder;PSR;25|) (SEQ G190 (COND ((NULL (COND ((SPADCALL (LETT |a| (SPADCALL |a| (QREFELT |$| 62)) |PSETCAT-;remainder;PSR;25|) (QREFELT |$| 42)) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |hRa| (|PSETCAT-;makeIrreducible!| (SPADCALL |a| |ps| (QREFELT |$| 76)) |$|) |PSETCAT-;remainder;PSR;25|) (LETT |a| (QCAR |hRa|) |PSETCAT-;remainder;PSR;25|) (LETT |r| (SPADCALL |r| (QCDR |hRa|) (QREFELT |$| 70)) |PSETCAT-;remainder;PSR;25|) (LETT |g| (SPADCALL |c| (LETT |lca| (SPADCALL |a| (QREFELT |$| 64)) |PSETCAT-;remainder;PSR;25|) (QREFELT |$| 65)) |PSETCAT-;remainder;PSR;25|) (LETT |b| (SPADCALL (SPADCALL (SPADCALL (QCDR |hRa|) (|PSETCAT-;exactQuo| |c| |g| |$|) (QREFELT |$| 70)) |b| (QREFELT |$| 66)) (SPADCALL (|PSETCAT-;exactQuo| |lca| |g| |$|) (SPADCALL |a| (QREFELT |$| 40)) (QREFELT |$| 67)) (QREFELT |$| 77)) |PSETCAT-;remainder;PSR;25|) (EXIT (LETT |c| |g| |PSETCAT-;remainder;PSR;25|))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (VECTOR |c| |b| |r|)))))))))) 

(DEFUN |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26| (|ps| |cs| |$|) (PROG (|p| |rs|) (RETURN (SEQ (COND ((SPADCALL |cs| (QREFELT |$| 80)) |ps|) ((SPADCALL |cs| (QREFELT |$| 81)) (LIST (|spadConstant| |$| 82))) ((QUOTE T) (SEQ (LETT |ps| (SPADCALL (ELT |$| 42) |ps| (QREFELT |$| 26)) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (EXIT (COND ((NULL |ps|) |ps|) ((SPADCALL (ELT |$| 24) |ps| (QREFELT |$| 43)) (LIST (|spadConstant| |$| 83))) ((QUOTE T) (SEQ (LETT |rs| NIL |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (SEQ G190 (COND ((NULL (COND ((NULL |ps|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |ps|) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (LETT |ps| (CDR |ps|) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (LETT |p| (QCAR (SPADCALL |p| |cs| (QREFELT |$| 76))) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (EXIT (COND ((NULL (SPADCALL |p| (QREFELT |$| 42))) (COND ((SPADCALL |p| (QREFELT |$| 24)) (SEQ (LETT |ps| NIL |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) (EXIT (LETT |rs| (LIST (|spadConstant| |$| 83)) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|)))) ((QUOTE T) (SEQ (SPADCALL |p| (QREFELT |$| 84)) (EXIT (LETT |rs| (CONS |p| |rs|) |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|))))))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (SPADCALL |rs| (QREFELT |$| 85)))))))))))))) 

(DEFUN |PSETCAT-;rewriteIdealWithRemainder;LSL;27| (|ps| |cs| |$|) (PROG (|p| |rs|) (RETURN (SEQ (COND ((SPADCALL |cs| (QREFELT |$| 80)) |ps|) ((SPADCALL |cs| (QREFELT |$| 81)) (LIST (|spadConstant| |$| 82))) ((QUOTE T) (SEQ (LETT |ps| (SPADCALL (ELT |$| 42) |ps| (QREFELT |$| 26)) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (EXIT (COND ((NULL |ps|) |ps|) ((SPADCALL (ELT |$| 24) |ps| (QREFELT |$| 43)) (LIST (|spadConstant| |$| 83))) ((QUOTE T) (SEQ (LETT |rs| NIL |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (SEQ G190 (COND ((NULL (COND ((NULL |ps|) (QUOTE NIL)) ((QUOTE T) (QUOTE T)))) (GO G191))) (SEQ (LETT |p| (|SPADfirst| |ps|) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (LETT |ps| (CDR |ps|) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (LETT |p| (QVELT (SPADCALL |p| |cs| (QREFELT |$| 87)) 1) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (EXIT (COND ((NULL (SPADCALL |p| (QREFELT |$| 42))) (COND ((SPADCALL |p| (QREFELT |$| 24)) (SEQ (LETT |ps| NIL |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) (EXIT (LETT |rs| (LIST (|spadConstant| |$| 83)) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|)))) ((QUOTE T) (LETT |rs| (CONS (SPADCALL |p| (QREFELT |$| 88)) |rs|) |PSETCAT-;rewriteIdealWithRemainder;LSL;27|))))))) NIL (GO G190) G191 (EXIT NIL)) (EXIT (SPADCALL |rs| (QREFELT |$| 85)))))))))))))) 

(DEFUN |PolynomialSetCategory&| (|#1| |#2| |#3| |#4| |#5|) (PROG (|DV$1| |DV$2| |DV$3| |DV$4| |DV$5| |dv$| |$| |pv$|) (RETURN (PROGN (LETT |DV$1| (|devaluate| |#1|) . #1=(|PolynomialSetCategory&|)) (LETT |DV$2| (|devaluate| |#2|) . #1#) (LETT |DV$3| (|devaluate| |#3|) . #1#) (LETT |DV$4| (|devaluate| |#4|) . #1#) (LETT |DV$5| (|devaluate| |#5|) . #1#) (LETT |dv$| (LIST (QUOTE |PolynomialSetCategory&|) |DV$1| |DV$2| |DV$3| |DV$4| |DV$5|) . #1#) (LETT |$| (GETREFV 90) . #1#) (QSETREFV |$| 0 |dv$|) (QSETREFV |$| 3 (LETT |pv$| (|buildPredVector| 0 0 (LIST (|HasCategory| |#2| (QUOTE (|IntegralDomain|))))) . #1#)) (|stuffDomainSlots| |$|) (QSETREFV |$| 6 |#1|) (QSETREFV |$| 7 |#2|) (QSETREFV |$| 8 |#3|) (QSETREFV |$| 9 |#4|) (QSETREFV |$| 10 |#5|) (COND ((|testBitVector| |pv$| 1) (PROGN (QSETREFV |$| 48 (CONS (|dispatchFunction| |PSETCAT-;roughUnitIdeal?;SB;16|) |$|)) (QSETREFV |$| 52 (CONS (|dispatchFunction| |PSETCAT-;roughBase?;SB;18|) |$|)) (QSETREFV |$| 54 (CONS (|dispatchFunction| |PSETCAT-;roughSubIdeal?;2SB;19|) |$|)) (QSETREFV |$| 57 (CONS (|dispatchFunction| |PSETCAT-;roughEqualIdeals?;2SB;20|) |$|))))) (COND ((|HasCategory| |#2| (QUOTE (|GcdDomain|))) (COND ((|HasCategory| |#4| (QUOTE (|ConvertibleTo| (|Symbol|)))) (PROGN (QSETREFV |$| 72 (CONS (|dispatchFunction| |PSETCAT-;headRemainder;PSR;23|) |$|)) (QSETREFV |$| 79 (CONS (|dispatchFunction| |PSETCAT-;remainder;PSR;25|) |$|)) (QSETREFV |$| 86 (CONS (|dispatchFunction| |PSETCAT-;rewriteIdealWithHeadRemainder;LSL;26|) |$|)) (QSETREFV |$| 89 (CONS (|dispatchFunction| |PSETCAT-;rewriteIdealWithRemainder;LSL;27|) |$|))))))) |$|)))) 

(MAKEPROP (QUOTE |PolynomialSetCategory&|) (QUOTE |infovec|) (LIST (QUOTE #(NIL NIL NIL NIL NIL NIL (|local| |#1|) (|local| |#2|) (|local| |#3|) (|local| |#4|) (|local| |#5|) (|List| 10) (0 . |members|) (|List| 9) (5 . |variables|) (|Boolean|) (10 . |<|) (|List| |$|) (16 . |concat|) (21 . |removeDuplicates|) (|Mapping| 15 9 9) (26 . |sort|) (32 . |mvar|) |PSETCAT-;variables;SL;4| (37 . |ground?|) (|Mapping| 15 10) (42 . |remove|) |PSETCAT-;mainVariables;SL;5| (48 . |=|) |PSETCAT-;mainVariable?;VarSetSB;6| (54 . |construct|) |PSETCAT-;collectUnder;SVarSetS;7| |PSETCAT-;collectUpper;SVarSetS;8| |PSETCAT-;collect;SVarSetS;9| (|Record| (|:| |under| |$|) (|:| |floor| |$|) (|:| |upper| |$|)) |PSETCAT-;sort;SVarSetR;10| (|Set| 10) (59 . |brace|) (64 . |=|) |PSETCAT-;=;2SB;11| (70 . |degree|) (75 . |<|) (81 . |zero?|) (86 . |any?|) (|Mapping| 15 10 10) (92 . |sort|) |PSETCAT-;triangular?;SB;14| |PSETCAT-;trivialIdeal?;SB;15| (98 . |roughUnitIdeal?|) (103 . |sup|) (109 . |+|) (115 . |=|) (121 . |roughBase?|) (126 . |rewriteIdealWithRemainder|) (132 . |roughSubIdeal?|) (138 . |=|) (144 . |roughSubIdeal?|) (150 . |roughEqualIdeals?|) (156 . |quo|) (|Union| |$| (QUOTE "failed")) (162 . |exquo|) (168 . |One|) (172 . |reductum|) (177 . |subtractIfCan|) (183 . |leadingCoefficient|) (188 . |gcd|) (194 . |*|) (200 . |monomial|) (206 . |*|) (212 . |-|) (218 . |*|) (|Record| (|:| |num| 10) (|:| |den| 7)) (224 . |headRemainder|) (230 . |gcd|) (236 . |one?|) (241 . |exactQuotient!|) (247 . |headRemainder|) (253 . |+|) (|Record| (|:| |rnum| 7) (|:| |polnum| 10) (|:| |den| 7)) (259 . |remainder|) (265 . |trivialIdeal?|) (270 . |roughUnitIdeal?|) (275 . |Zero|) (279 . |One|) (283 . |primitivePart!|) (288 . |removeDuplicates|) (293 . |rewriteIdealWithHeadRemainder|) (299 . |remainder|) (305 . |unitCanonical|) (310 . |rewriteIdealWithRemainder|))) (QUOTE #(|variables| 316 |trivialIdeal?| 321 |triangular?| 326 |sort| 331 |roughUnitIdeal?| 337 |roughSubIdeal?| 342 |roughEqualIdeals?| 348 |roughBase?| 354 |rewriteIdealWithRemainder| 359 |rewriteIdealWithHeadRemainder| 365 |remainder| 371 |mainVariables| 377 |mainVariable?| 382 |headRemainder| 388 |collectUpper| 394 |collectUnder| 400 |collect| 406 |=| 412)) (QUOTE NIL) (CONS (|makeByteWordVec2| 1 (QUOTE NIL)) (CONS (QUOTE #()) (CONS (QUOTE #()) (|makeByteWordVec2| 89 (QUOTE (1 6 11 0 12 1 10 13 0 14 2 9 15 0 0 16 1 13 0 17 18 1 13 0 0 19 2 13 0 20 0 21 1 10 9 0 22 1 10 15 0 24 2 11 0 25 0 26 2 9 15 0 0 28 1 6 0 11 30 1 36 0 11 37 2 36 15 0 0 38 1 10 8 0 40 2 8 15 0 0 41 1 10 15 0 42 2 11 15 25 0 43 2 11 0 44 0 45 1 0 15 0 48 2 8 0 0 0 49 2 8 0 0 0 50 2 8 15 0 0 51 1 0 15 0 52 2 6 11 11 0 53 2 0 15 0 0 54 2 6 15 0 0 55 2 6 15 0 0 56 2 0 15 0 0 57 2 7 0 0 0 58 2 7 59 0 0 60 0 7 0 61 1 10 0 0 62 2 8 59 0 0 63 1 10 7 0 64 2 7 0 0 0 65 2 10 0 7 0 66 2 10 0 7 8 67 2 10 0 0 0 68 2 10 0 0 0 69 2 7 0 0 0 70 2 0 71 10 0 72 2 10 7 7 0 73 1 7 15 0 74 2 10 0 0 7 75 2 6 71 10 0 76 2 10 0 0 0 77 2 0 78 10 0 79 1 6 15 0 80 1 6 15 0 81 0 10 0 82 0 10 0 83 1 10 0 0 84 1 11 0 0 85 2 0 11 11 0 86 2 6 78 10 0 87 1 10 0 0 88 2 0 11 11 0 89 1 0 13 0 23 1 0 15 0 47 1 0 15 0 46 2 0 34 0 9 35 1 0 15 0 48 2 0 15 0 0 54 2 0 15 0 0 57 1 0 15 0 52 2 0 11 11 0 89 2 0 11 11 0 86 2 0 78 10 0 79 1 0 13 0 27 2 0 15 9 0 29 2 0 71 10 0 72 2 0 0 0 9 32 2 0 0 0 9 31 2 0 0 0 9 33 2 0 15 0 0 39)))))) (QUOTE |lookupComplete|))) 
@
\section{domain GPOLSET GeneralPolynomialSet}
<<domain GPOLSET GeneralPolynomialSet>>=
)abbrev domain GPOLSET GeneralPolynomialSet
++ Author: Marc Moreno Maza
++ Date Created: 04/26/1994
++ Date Last Updated: 12/15/1998
++ Basic Functions:
++ Related Constructors:
++ Also See: 
++ AMS Classifications:
++ Keywords: polynomial, multivariate, ordered variables set
++ References:
++ Description: A domain for polynomial sets.
++ Version: 1

GeneralPolynomialSet(R,E,VarSet,P) : Exports == Implementation where

  R:Ring
  VarSet:OrderedSet
  E:OrderedAbelianMonoidSup
  P:RecursivePolynomialCategory(R,E,VarSet)
  LP ==> List P
  PtoP ==> P -> P

  Exports ==  PolynomialSetCategory(R,E,VarSet,P)  with

     convert : LP -> $
       ++ \axiom{convert(lp)} returns the polynomial set whose members 
       ++ are the polynomials of \axiom{lp}.

     finiteAggregate
     shallowlyMutable

  Implementation == add

     Rep := List P

     construct lp ==
       (removeDuplicates(lp)$List(P))::$

     copy ps ==
       construct(copy(members(ps)$$)$LP)$$

     empty() ==
       []

     parts ps ==
       ps pretend LP

     map (f : PtoP, ps : $) : $ ==
       construct(map(f,members(ps))$LP)$$

     map! (f : PtoP, ps : $) : $  ==
       construct(map!(f,members(ps))$LP)$$

     member? (p,ps) ==
       member?(p,members(ps))$LP

     ps1 = ps2 ==
       {p for p in parts(ps1)} =$(Set P) {p for p in parts(ps2)}

     coerce(ps:$) : OutputForm ==
       lp : List(P) := sort(infRittWu?,members(ps))$(List P)
       brace([p::OutputForm for p in lp]$List(OutputForm))$OutputForm

     mvar ps ==
       empty? ps => error"Error from GPOLSET in mvar : #1 is empty"
       lv : List VarSet := variables(ps)
       empty? lv => error"Error from GPOLSET in mvar : every polynomial in #1 is constant"
       reduce(max,lv)$(List VarSet)

     retractIfCan(lp) ==
       (construct(lp))::Union($,"failed")

     coerce(ps:$) : (List P) ==
       ps pretend (List P)

     convert(lp:LP) : $ ==
       construct lp

@
\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 PSETCAT PolynomialSetCategory>>
<<domain GPOLSET GeneralPolynomialSet>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}