Abstract. A semantic formulation of Lindley and Stark’s ⊤⊤-lifting is given. We first illustrate our semantic formulation of the ⊤⊤-lifting in Set with several examples, and apply it to the logical predicates for Moggi’s computational metalanguage. We then abstract the semantic ⊤⊤-lifting as the lifting of strong monads across bifibrations with lifted symmetric monoidal closed structures. 1

this paper, but for the logical relations, which are fixed arity relations as defined above in case of [9]. It is an open problem whether the characterization holds at higher types

An arithmetical proof of the strong normalization

An arithmetical proof of the strong normalization for the λ-calculus with recursive equations on types

In this paper, we give the first relationally parametric model of the (extensional) calculus of constructions. Our model remains as simple as traditional PER models of dependent types, but unlike them, our model additionally permits relating terms at different implementation types. Using this model, we can validate the soundness of quotient types, as well as derive strong equality axioms for Church-encoded data, such as the eta-law for strong dependent pair types. 1.