Is there a good software that helps with the outline of a proof?

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Since I'm self-studying, it'll be very helpful to have a software to check the correctness of my proofs. Is there a software like that? And more importantly, is it a good idea to use such a software?

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Such a software, in the sense that you have in mind, doesn't exist. There are formal proof checkers but 'formal proofs' look nothing like the proofs you would write and like to be checked.

Also, I don't think it's a good idea to use such tools for practice purposes, if they existed . Instead of relying on external sources, carefully think through every step of your proof. This not only allows you to self-verify your work but in addition builds a certain mathematical maturity that is crucial for any kind of mathematical work.