CoqĬollected here are various resources for working (and playing) with HoTT in Coq, mostly hosted at GitHub. One of the main directions of current work in HoTT is the development of an infrastructure that will allow to formally certify new features for their compatibility with the standard model. The HoTT library took the opposite approach of using freely all of the features of the CIC and adding to them more features that can be broadly described as experiments with various approaches to higher inductive types. To circumvent this problem while waiting for the changes in the universe management that will allow to use Resizing Rules the UniMath library uses type-in-type patch that is expected to become an off-the-shelf option in Coq 8.5. This however makes it impossible to carry through some important constructions such as the set quotients of types without running into an increase of complexity due to the way the universes are managed in Coq. The UniMath library uses only the MLTT fragment of the CIC (with some exceptions in the Ktheory library). This situation led to the development of two different approaches to univalent formalization of mathematics in Coq and Agda. The question of whether a standard univalent model can be extended to all of the CIC is open and somewhat fluid since the the type theory that is implemented in Coq and in even greater degree in Agda is changing from one release to another. The standard univalent model that allows one to use MLTT to formalize abstract mathematics in the univalent style has been informally checked to extend to some of the subsets of CIC. The Martin-Lof type theory can be seen as a fragment of CIC. An intensional dependent type theory called the Calculus of Inductive Constructions (CIC) has implementations in proof assistants such as Coq and Agda.
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