We study the λΠΣS ≤ calculus, which contains singleton types S(M) classifying terms of base type provably equivalent to the term M. The system includes dependent types for pairs and functions (Σ and Π) and a subtyping relation induced by regarding singletons as subtypes of the base type. The decidability of type checking for this language is non-obvious, since to type check we must be able to determine equivalence of well-formed terms. But in the presence of singleton types, the provability of an equivalence judgment Γ ⊢ M1 ≡ M2: A can depend both on the typing context Γ and on the particular type A at which M1 and M2 are compared. We show how to prove decidability of term equivalence, hence of type checking, in λΠΣS ≤ by exhibiting a type-directed algorithm for directly computing normal forms. The correctness of normalization is shown using an unusual variant of Kripke logical relations organized around sets; rather than defining a logical equivalence relation, we work directly with (subsets of) the corresponding equivalence classes. We then provide a more efficient algorithm for checking type equivalence without constructing normal forms. We also show that type checking, subtyping, and all other judgments of the system are decidable.

We present a system of refinement types for LF in the style of recent formulations where only canonical forms are well-typed. Both the usual LF rules and the rules for type refinements are bidirectional, leading to a straightforward proof of decidability of typechecking even in the presence of intersection types. Because we insist on canonical forms, structural rules for subtyping can now be derived rather than being assumed as primitive. We illustrate the expressive power of our system with several examples in the domain of logics and programming languages.

The logical framework LF and its implementation as the Twelf metalogic provide both a practical system and a proven methodology for representing deductive systems and their metatheory in a machine-checkable way. An extension of LF with refinement types provides a convenient means for representing certain kinds of judgemental inclusions in an intrinsic manner. I propose to carry out such an extension in full, adapting as much of the Twelf metatheory engine as possible to the new system, and I intend to argue that the extension is both useful and practical. 1

views and conclusions contained in this document are those of the author, and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government, or any other entity. Keywords: refinement types, logical frameworks, subtyping, intersection types, The logical framework LF and its metalogic Twelf can be used to encode and reason about a wide variety of logics, languages, and other deductive systems in a formal, machine-checkable way. Recent studies have shown that ML-like languages can profitably be extended with a notion of subtyping called refinement types. A refinement type discipline uses an extra layer of term classification above the usual type system to more accurately capture certain properties of terms. I propose that adding refinement types to LF is both useful and practical. To support the claim, I exhibit an extension of LF with refinement types called LFR, work out important details of its metatheory, delineate a practical algorithm for refinement type reconstruction, and present several case studies