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coq
Coq is a formal proof management system. It provides a formal language to write mathematical definitions, executable algorithms and theorems together with an environment for semi-interactive development of machine-checked proofs.
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If you're interested in learning about how to formalize proofs, you can to a certain extent hand-wave a formal argument of correctness or validity, but for a serious treatment, in the 21st century, you should be using a formal proof assistant such as Coq or Lean, that is to say, a computer program specifically written for the express purpose of formalizing mathematical proofs.
If you're interested in learning about how to formalize proofs, you can to a certain extent hand-wave a formal argument of correctness or validity, but for a serious treatment, in the 21st century, you should be using a formal proof assistant such as Coq or Lean, that is to say, a computer program specifically written for the express purpose of formalizing mathematical proofs.