Is %i an integer? - Adding more facts to the database

There seem to be two issues, perhaps confused in Maxima, apparently in 
common lisp...

1. The data layout used for storing a value
2. The mathematical domain to which the value belongs.

For example,
(a) 3 is an integer and can be stored in a box for integers.
(b) 3 can also be stored in a box for complex numbers, with imaginary 
part 0.
(c) 3 can be stored in a box for rationals (numerator 3, denominator 1),

In some situations (b) and (c) will be coerced to be (a).
In common lisp,  (complex 3 0)  returns 3.  (complex 3 0.0) returns 
#c(3.0 0.0)
(/ 3 1) returns 3.

(complexp (complex 3 0.0))   returns t
(complexp (complex 3 0))   returns nil.

But the predicate rationalp  does not require that the box be a 
"rational: box since
(rationalp 3) returns t

...................
2.  From the math perspective it seems that we have another set of 
questions to be asked.

Is a symbol restricted to having values in a particular domain?
Is an explicit constant a member of a particular domain?

 From this perspective, integers are a subset of  rational are a subset 
of real  [and  also a subset of complex, with imaginary part = 0]
Since the complex numbers are not ordered,   assume(a<b) means that a, b 
are real.
At least for it to be true.  Do we want to say a<b is false because a,b 
are complex??
Also someone could use "<" for some other ordering, not unusual in 
mathematical discourse.

Similarly assume(evenp(x)) probably means that x is an integer.  Again, 
do we want to say that 3.14 is "not even"?

There are lots of other possible domains and towers of domains, as well 
as features.

RJF


though perhaps Robert Dodier wrote:
> On 6/28/09, Dieter Kaiser <drdieterkaiser at web.de> wrote:
>
>   
>> I have added the following facts to the database:
>>
>>    (kind $%i $imaginary)
>>    (kind $%pi $real)
>>    (kind $%gamma $real)
>>    (kind $%phi $real)
>>     
>
> OK by me. But in addition perhaps %pi, %gamma, and %phi
> should be declared irrational. (Or maybe transcendental for
> %pi and %gamma, but I don't remember if featurep and kindp
> handle that correctly.)
>
>   
>> Next I have improved the function NONINTEGERP.
>>
>> ;	  ((atom e) (kindp e '$noninteger))
>>           ((atom e) (or (kindp e '$noninteger)
>>                         (kindp e '$rational)
>>                         (kindp e '$real)
>>                         (kindp e '$complex)))
>>     
>
> I don't think this is right. NONINTEGERP should return T when
> E is demonstrably noninteger. Some rationals, reals, and
> complex numbers are integers, right? So it can't use those
> declarations to determine noninteger-ness.
>
> I think testing for noninteger, irrational, imaginary, and
> transcendental would be OK.
>
> Thanks for working on this problem. I think handling declarations
> correctly is very important.
>
> best
>
> Robert Dodier
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