From Rewrite Rules to Axioms in the \(\lambda \varPi \) -Calculus Modulo Theory

The \(\lambda \varPi \) -calculus modulo theory is an extension of simply typed \(\lambda \) -calculus with dependent types and user-defined rewrite rules. We show that it is possible to replace the rewrite rules of a theory of the \(\lambda \varPi \) -calculus modulo theory by equational axioms, when this theory features the notions of proposition and proof, while maintaining the same expressiveness. To do so, we introduce in the target theory a heterogeneous equality, and we build a translation that replaces each use of the conversion rule by the insertion of a transport. At the end, the theory with rewrite rules is a conservative extension of the theory with axioms.