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Looking at Gregory and Robert’s "matchers" link, which also covers "assertThat", it seems to provide a library of matcher assertion objects. E.g. instead of this TestCase-coupled code:
self.assertEquals(something, "expected value")
you would write
from testtools.matchers import Equals
self.assertThat(something, Equals("expected value"))
Implementing a custom matcher (say HasAttr for Serhiy’s first use case) seems to involve creating the HasAttr class with a match() method, and maybe another HasAttrMismatch class (though maybe you could get away with just a shared generic mismatch class?).
It seems to me that if you ignore the vaguely-documented TestCase.failureException attribute, you can fairly easily decouple the existing methods from a TestCase instance:
def assert_equal(first, second, msg=None):
 # Implementation could be copied independently of any test state
 TestCase().assertEqual(first, second, msg)
The matcher scheme would be more flexible though, because you can compose matcher objects.