Internalization of extensional equality

Abstract

We give a type system in which the universe of types is closed by reflection into it of the logical relation defined externally by induction on the structure of types. This contribution is placed in the context of the search for a natural, syntactic construction of the extensional equality type (Tait [1995], Altenkirch [1999], Coquand [2011], Licata and Harper [2012], Martin-Lof [2013]). The system is presented as an extension of lambda-*, the terminal pure type system in which the universe of all types is a type. The universe inconsistency is then removed by the usual method of stratification into levels. We give a set-theoretic model for the stratified system. We conjecture that Strong Normalization holds as well.