On cubical homotopy type theory implemented in the proof assistant Cubical Agda:
Andrea Vezzosi, Anders Mörtberg and Andreas Abel, Cubical Agda: A Dependently Typed Programming Language with Univalence and Higher Inductive Types, 2019 (pdf)
Anders Mörtberg, Cubical methods in HoTT/UF, 2019 (pdf)
Exposition in view of synthetic homotopy theory:
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