We present the first verified normalization-by-evaluation algorithm for System F ω, the simplest impredicative type theory with computation on the type level. Types appear in three shapes: As syntactical types, as type values which direct the reification process, and as semantical types, i.e., sets of total values. The three shapes are captured by the new concept of a type structure, and the fundamental theorem now states that an induced structure is a type substructure. This work is an attempt at an algebraic treatment of type theory based on typed applicative structures rather than categories. 1

We present a procedure for computing normal forms of terms in Abadi and Cardelli’s functional object calculus. Even when equipped with simple types, terms of this calculus are not terminating in general, and we draw on recent ideas about the normalization by evaluation paradigm for the untyped lambda calculus. Technically, we work in the framework of Shinwell and Pitts ’ FM-domain theory, which leads to a normalization procedure for the object calculus that is directly implementable in a language like Fresh O’Caml.