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This study provides some results about two-level type-theoretic notions in away that the proofs are fully formalizable in a proof assistant implementingtwo-level type theory such as Agda. The difference from prior works is thatthese proofs do not assume any abuse of notation, providing us with more directformalization. Moreover, some new notions, such as function extensionality forcofibrant types, are introduced. The necessity of such notions arises duringthe task of formalization. In addition, we provide some novel results aboutinductive types using cofibrant exo-nat, the natural number type at thenon-fibrant level. While emphasizing the necessity of this axiom by citing newapplications as justifications, we also touch upon the semantic aspect of thetheory by presenting various models that satisfy this axiom.

Formalizing two-level type theory with cofibrant exo-nat