[Axiom-math] Re: Type equivalence of domains in Axiom and Aldor

[Francois Maltey]

Dear Martin, 

> what does a looping construct have to do with a Cartesian product?

The construct arround reduce/concat isn't so usual in order to 
built several overlapped loops.
cartesian product and repeat keyword are two possibilities.

> > The * isn't the better operator because this locks
> >     {1,2,3} * {1,2,3} = {1,2,3,4,6,9} 
> > and {1,2,3} + {1,2,3} = {2,3,4,5,6}
> 
> I actually wondered already once, why "+" is not union$Set, "-" not
> difference$Set.  

I think that operator for sets are missing. 
union (union (E, F), G) is less clever than E union F union G.

But why do you want a +. It might be possible to have more operators 
than only +, -, *, /, ^, ::, $, # and @.

> I didn't think of the possibility above, although I doubt that
> it would be too useful.  

For finite set it's not useful, 
but very useful for intervals ; and intervals and set are close. 

I like to use it for chaotic systems, with numerical instability.
or in order to separate root of a function.

newton methods can forget some mutiple roots.
interval methods, with operations over intervals, can add false roots. 

> After all, you can even define it only if the elements
> understand "+" and "*". 

I meet this needs for functions. 
axiom doesn't understand that sin+cos is the fuction x +-> (sin x + cos x)

François