Extensional equivalence and singleton types

We study the &lamda;^ΠΣ<i>S</i>_≤ calculus, which contains singleton types <i>S</i>(<i>M</i>) classifying terms of base type provably equivalent to the term <i>M</i>. 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 ⊢ <i>M</i>₁ &eqiv;<i>M</i>₂ : <i>A</i> can depend both on the typing context Γ and on the particular type <i>A</i> at which <i>M</i>₁ and <i>M</i>₂ are compared.We show how to prove decidability of term equivalence, hence of type checking, in &lamda;^ΠΣ<i>S</i>_≤ 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.The &lamda;^ΠΣ<i>S</i>_≤ calculus models type constructors and kinds in the intermediate language used by the TILT compiler for Standard ML to implement the SML module system. The decidability of &lamda;^ΠΣ<i>S</i>_≤ term equivalence allows us to show decidability of type checking for TILT's intermediate language. We also obtain a consistency result that allows us to prove type safety for the intermediate language. The algorithms derived here form the core of the type checker used for internal type checking in TILT.