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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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That's untrue. There are type-systems and type-checkers which can check for correctness, as specified by a mathematical formula. This is proof assistants such as Coq[1] work. For example, take a look at fiat-crypto [2], a project that generates code that is correct by mathematical proof, and is now deployed in Chrome. Another famous example is the CompCert [3] compiler, a C compiler along with a correctness proof that the compiler generates assembly which correctly simulates the C semantics.
[1] https://coq.inria.fr/