Kuratowski finite sets in the UniMath Library

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This paper focuses on implementing and verifying the proofs presented in ``Finite Sets in Homotopy Type Theory" within the UniMath library. The UniMath library currently lacks support for higher inductive types, which are crucial for reasoning about finite sets in Homotopy Type Theory. This paper addresses that issue and introduces higher inductive types to UniMath. This is used to develop a computer-checked implementation of the proofs within "Finite Sets in Homotopy Type Theory." This implementation enables future research on finite sets in HoTT by providing accessible and reliable proofs.

This paper defines finite sets as Kuratowski-finite. This is in contrast with the most common notion of finiteness, e.g. Bishop-finite and enumerated types. I argue that Kuratowski-finiteness is the most general finite for which the usual operations of finite types and sub-objects can be operated upon.