Computational Paths and Identity Types

We introduce a new way of formalizing the intensional identity type based on the notion of computational paths which will be taken to be proofs of propositional equality, and thus terms of the identity type. Our approach results in an elimination rule different than the one given by Martin-Löf in his intensional identity type. In order to show the validity and power of our approach, we formulate and prove the basic concepts, lemmas and theorems of Homotopy Type Theory using computational paths. We also show that these proves and formulations resulted, as a side effect, in the improvement, by means of the addition of new rules, of a term rewrite system known as LNDEQ − TRS, originally proposed by de Oliveira (1995).