[aureianimus]

In a lot of cases you can get far by locally proofreading the definitions.

Trying to formally prove something and then failing is a common way people find out they forgot to add an hypothesis.

Another pitfall is defining some object, but messing up the definitions, such that there's actually no object of that kind. This is addressed by using test objects. So suppose you define what a ring is, then you also prove that real numbers and polynomials are examples of the thing you defined.