RE: [Axiom-developer] Question concerning types...

[Bill Page]

On September 18, 2006 1:50 PM C Y wrote:
> 
> Sorry in advance if this question is a bit daft...
>

Not at all. There is not such thing as a daft question -
however always keep in mind that the same is not true of
answers. :-)
 
> --- Bill Page wrote:
> 
> > I used '@POLY(INT)' because I wanted to tell the Axiom interpreter
> > not to look of other coercions which would have made the
> > interpretation of '/' possible but no longer in 'POLY INT'.
> > 
> > Should we say that 'a1' represents an indefinite integer?
> > Tentatively, I think so.
> 
> Does this mean that an indefinite integer couldn't be substituted in
> anywhere a "definite" integer could be used?
> 

No. The issue here is that what we want to call "indefinite integer"
has to come from some domain in Axiom. My proposal is that to define
this in algebraic terms what we need is a domain like Polynomial which
consists of some symbols and expressions (of a specific kind) over
these symobls. So it is clear, right? that the type

  Polynomial Integer

consists of a large clase of expressiions of that type. And saying

  a1:Polynomial Integer

is just a way of saying that the variable a1 will take values from
this domain.

But because the coefficients of the polynomial must come from the
domain Integer we know that these Integers are embedded in this class
of expressions as polynomials of degree 0, so we have no problem
specifying that a certain variable suchs as 'a1' is exact such an
integer (polynomial of degree 0).

In general both Integer and Polynomial Integer has Ring so, yes
it is true that Polynomial Integer can be used in (most) places
where would like to use Integer (but were no specific value is
required).

The only concern I have is whether or not Polynomial Integer is
"big enough" to model everything that we would want to mean by
"indefinite integer".

Regards,
Bill Page.