The Theory of an Arbitrary Higher $\lambda$-Model

One takes advantage of some basic properties of every homotopic $\lambda$-model (e.g.\ extensional Kan complex) to explore the higher $\beta \eta$-conversions, which would correspond to proofs of equality between terms of a theory of equality of any extensional Kan complex. Besides, Identity types based on computational paths are adapted to a type-free theory with higher $\lambda$-terms, whose equality rules would be contained in the theory of any $\lambda$-homotopic model.