Re: if-expression and variables

Dear Stefan:

When you input "x < 10" into a symbolic computation 
program, the meaning is at first ambiguous. In the context 
of a function such as test1, a meaning can be obtained by 
assuming x is of the same type as 10 (or at least numeric) 
and the usual ordering of numbers is used for the "<" 
operation. However, if the input x is not numeric, such as 
the symbol y you use, then it does not make sense to 
compare a symbol y with a number, except in the context of 
polynomials (y and 10 both treated as polynomials in y). 
There, the usual ordering is an extension of a 
term-ordering and y > 10, as polynomials, and that is why 
you get 5y^2.  (BTW, you have a typo in your definition of 
test1.)

To implement what you expect for "test1 y" would require a 
whole new setup, treating "y < 10" as an unevaluated 
expression (hold the evaluation until more information is 
known about y). But even if that is the setup, "test1 y" 
could only return "test1 y", unless there is an 
instruction from the user to "expand" the definition of 
test1.

So the answer to your question "How is it possible ..." 
is, "It is not possible, in the context of piecewise 
functions." because no matter what the system does, you 
gain no new information other than what you inputted, 
until you let the system know what y is, numerically.

At a higher level of symbolic computation, it is possible 
to simplify a boolean expression of inequalities (in 
Axiom, as far as I know, there is no such domain yet; 
Mathematica has a function called "Reduce" that does 
simplifications involving inequalities and equations). 
However, that would not be in the simple context of a 
piecewise function.

Tim Daly has done some experiments with a proviso domain 
in Axiom. He may be able to tell you more.

William


On Sat, 30 Apr 2011 20:40:28 +0200
  Stefan Karrmann <[email protected]> wrote:
> Dear all,
> 
> I'm new to axiom and have a problem with piecewise 
>functions.
> 
> test1 (x | x < 10) == 2*x
> test1 (x | x < 10) == 5*x^2
> test1
> -> 
>   test1 (x | x < 10) == 2x
>   test1 (x | ^ x < 10) == 5x
>                                                   Type: 
>FunctionCalled
> test1 y
> ->
>     2
>   5y
> 
> I expected something like (if y < 10 then 2*y else 
>5*y**2).
> 
> How is it possible to pass a Variable to a piecewise 
>function respecting
> the pieces?
> 
> PS: Using a block and => or explicit if-then-else does 
>not help.
> 
> -- 
> Kind regards,
> Stefan
> 
> 
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William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/