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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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InfluxDB
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Sure we can. That's a common way to do it if your foundational theory is some version of type theory. For example, when you want to talk about sets in the Coq proof assistant (wich uses type theory as its foundational theory), this is exactly what you would do. Your "set of values of type T" would be something of the type T → Prop.
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How are people like Andrew Wiles and Grigori Perelman able to work on popular problems for years without others/the research community discovering the same breakthroughs? Is it just luck?