How to determine if a regex is orthogonal to another regex?

Question

I guess my question is best explained with an (simplified) example.

Regex 1:

^\\d+_[a-z]+$

Regex 2:

^\\d*$
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Answer 1

Proving that one regular expression is orthogonal to another can be trivial in some cases, such as mutually exclusive character groups in the same locations. For any but the simplest regular expressions this is a nontrivial problem. For serious expressions, with groups and backreferences, I would go so far as to say that this may be impossible.