Determining equality of symbolic expressions

[adasher]

Steps To Reproduce

var('x, y, n')
assume(n,'integer')
assume(n >= 0)
bool((x-y)^n==(-1)^n*(y-x)^n)

Expected Behavior

Returns True.

Actual Behavior

Returns False.

Additional Information

https://ask.sagemath.org/question/80970/how-to-determine-equality-of-symbolic-expressions/

Environment

• OS: Ubuntu 22.04
• Sage Version: 9.5

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[orlitzky]

There are two issues here:

1. Calling bool() on symbolic equalities is a longstanding pain point in our user interface. There are three possible return values: True, False, and "I don't know." For compatibility with python (IIRC) we make sure to always return either True or False... but then what happens to "I don't know"? It gets turned in to False! So you never really know whether or not your equation was provably false, or just too hard to decide.
2. Sage isn't smart enough to apply your assumptions to this equation, so it doesn't know, for example, that n isn't 1/2. It therefore cowardly refuses to simplify what might be a multi-valued square root to the single value that would make this equation true.

To work around the second problem, subtract the left-hand side of your equation from the right-hand side, and call canonicalize_radical() on the result. It should come out to zero.

[maxale]

@orlitzky: An equivalent comparison via ((x-y)^n - (-1)^n*(y-x)^n).is_zero() is not bound to always return True or False, and could tell us "Don't know" (say, in the form of an exception), but it still returns wrong False.

Related: #38586

[orlitzky]

I don't think it would be too terrible to throw an exception in bool(). No one likes the existing behavior and I'm certainly not defending it; the main roadblock is how much work it would be to change it in a consistent way. A few people have started to work on it over the years but no one has made much progress, e.g.

• wrong symbolic results in case the answer is not known #17700
• symbolic relations metaticket #19162