On Wed, Jun 17, 2026 at 05:20:27PM +0800, Qian Yun wrote:
> x1:=(y-sqrt(y^2))/(x-sqrt(x^2))
> x2:=1/(y+sqrt(y^2))*(x+sqrt(x^2))
> (x1=x2)$EXPR(INT)
> x1-x2
>
>
> So (x1=x2) was false before my recent commit 7af446cb,
> afterwards it becomes true.
>
> I think that is acceptable.
>
> However (x1-x2) is not zero, before/after this commit.
>
> That's caused by the '/' in 'reduc', common gcd factor
> is removed from numerator on the first loop.
>
> Shall we try to fix it?
>
> Or reject it because it contains dependent algebraic kernels?
Dependent kernels mean that we do not have a field, but basic
things should still work properly. More precisly, sqrt(x^2)
means extention by element s such that s^2 = x^2. This is
well defined and gives ring with zero divisors. For such
ring we may consider full ring of fractions, that is set
of fractions n/d such that d is not a zero divisor. x1
and x2 above are problematic because in both cases denominator
is a zero divisor. Both x1 and x2 make sense if we restrict
branches of square root so that denominators are non zero.
If we do that both for x1 and x2, then we get value zero
for both. So possible result of x1 = x2 could be
[true | x = -sqrt(x^2) and y = sqrt(y^2)]
and for x1 - x2
[0 | x = -sqrt(x^2) and y = sqrt(y^2)]
where I used '[a | b]' to denote that result a is valid under
condition b.
Without conditional expressions (which we do not have ATM) we
could try to reject x1 and x2. This is expensive but doable in
algebraic cases. But if we have logarithms and want to reason
about branches of logarithms, then problem in general is
undecidable (with conditional expressions we could return
problematic conditions to the user, without we need to decide).
Rejecting x1 and x2 is also problematic because we could create
package(s) which do something sensible about branches. If
x1 and x2 were rejected we could not create them as expressions
and pass to such a package, so we would need a new domain
allowing x1 and x2.
Currently probably handling of x1 and x2 does not matter too much,
as current Expression is not able to give sensible result for x1 = x2
or x1 - x2. But rejecting them is also rather unattractive.
--
Waldek Hebisch
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