Generic derivation of induction for impredicative encodings in Cedille

Abstract

This paper presents generic derivations of induction for impredicatively typed lambda-encoded datatypes, in the Cedille type theory. Cedille is a pure type theory extending the Curry-style Calculus of Constructions with implicit products, primitive heterogeneous equality, and dependent intersections. All data erase to pure lambda terms, and there is no built-in notion of datatype. The derivations are generic in the sense that we derive induction for any datatype which arises as the least fixed point of a signature functor. We consider Church-style and Mendler-style lambda-encodings. Moreover, the isomorphism of these encodings is proved. Also, we formalize Lambek's lemma as a consequence of expected laws of cancellation, reflection, and fusion.