Re: [Axiom-developer] Eval in Axiom/Aldor

[Ralf Hemmecke]

On 11/15/2006 07:44 AM, Page, Bill wrote:
> On Tuesday, November 14, 2006 6:20 PM Antoine Hersen wrote:
> > I get nasty FOAM run time error with Ralf solution (I used it
> > in a more involved context)
>
> Ralf wrote:
>
> > Replace "eval(a, x, 1)" by
> >
> >     z: SingletonAsOrderedSet := create();
> >     eval(numer a, z, 1)/eval(denom a, z, 1);
>
> I don't understand why Ralf introduces a new variable z. But
> I think that is the right idea.

Since I worked backwards to find where "eval" is implemented. That 
compiled and I was in a rush. I don't claim it is the way to go.

[snip]

> However
>
>   UnivariatePolynomial(x, F) has
> InnerEvalable(UnivariatePolynomial(x,F),
>   UnivariatePolynomial(x,F))
>
> is true so one way that does work is:
>
>     import from UnivariatePolynomial(x, F);
>     import from NonNegativeInteger;
>
>     test1(a:Fraction UnivariatePolynomial(x,F)):Fraction
> UnivariatePolynomial(x,F) == {
>         eval(numer a, monomial(1$F,1),1)/eval(denom a,
> monomial(1$F,1),1)
>     };

> monomial(1$F,1) is just one way to get x in the right form.
> (Perhaps Ralf missed this possibility.)

I hope you see that the eval from above is not the same function as the 
eval here.

I considered the definitions ...

UnivariatePolynomial(x:Symbol, R:Ring):
  UnivariatePolynomialCategory(R) with ...

UnivariatePolynomialCategory(R:Ring): Category ==
 Join(PolynomialCategory(R, NonNegativeInteger, SingletonAsOrderedSet),
      Eltable(R, R), Eltable(%, %), DifferentialRing,
      DifferentialExtension R) with ...

PolynomialCategory(R:Ring, E:OrderedAbelianMonoidSup, VarSet:OrderedSet):
        Category ==
  Join(PartialDifferentialRing VarSet, FiniteAbelianMonoidRing(R, E),
       Evalable %, InnerEvalable(VarSet, R),
       InnerEvalable(VarSet, %), RetractableTo VarSet,
       FullyLinearlyExplicitRingOver R) with

while I am using InnerEvalable(VarSet, R) (which has nothing to do with 
the symbol x from UnivariatePolynomial, you suggested

InnerEvalable(%, %)

from UnivariatePolynomial(x,F) where I must say, that I would have to 
dig deeper in the category hierarchy of UP in order to find that signature.

Two different "eval"s, where yours is probably a bit more general. I 
don't believe that they end up in the same implementation.

Ralf