I think unit tests serve these purposes:

- Document an API by example. (We do that in doctests.)

- Guard against regression. (Normal tests.)

- Test emergent properties, i.e. those that cannot be trivially deduced from reading the source code. (Doctest or normal test, depending on whether the test is also documentation-worthy or not.)

They can't make 100% sure that the code is valid, they can only increase confidence. And since we're in a grey area, we can weigh: Is the added effort to run and possibly maintain a unit test worth the increased confidence in the code's validity?

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