  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.

Logic in Computer Science