Intensional polymorphism, the ability to dispatch to different routines based on types at run time, enables a variety of advanced implementation techniques for polymorphic languages, including tag-free garbage collection, unboxed function arguments, polymorphic marshalling, and flattened data structures. To date, languages that support intensional polymorphism have required a type-passing (as opposed to type-erasure) interpretation where types are constructed and passed to polymorphic functions at run time. Unfortunately, type-passing suffers from a number of drawbacks: it requires duplication of constructs at the term and type levels, it prevents abstraction, and it severely complicates polymorphic closure conversion.We present a type-theoretic framework that supports intensional polymorphism, but avoids many of the disadvantages of type passing. In our approach, run-time type information is represented by ordinary terms. This avoids the duplication problem, allows us to recover abstraction, and avoids complications with closure conversion. In addition, our type system provides another improvement in expressiveness; it allows unknown types to be refined in place thereby avoiding certain beta-expansions required by other frameworks.

Machine-checked proofs of properties of programming languages have become a critical need, both for increased confidence in large and complex designs and as a foundation for technologies such as proof-carrying code. However, constructing these proofs remains a black art, involving many choices in the formulation of definitions and theorems that make a huge cumulative difference in the difficulty of carrying out large formal developments. The representation and manipulation of terms with variable binding is a key issue.We propose a novel style for formalizing metatheory, combining locally nameless representation of terms and cofinite quantification of free variable names in inductive definitions of relations on terms (typing, reduction, . . . ). The key technical insight is that our use of cofinite quantification obviates the need for reasoning about equivariance (the fact that free names can be renamed in derivations); in particular, the structural induction principles of relations defined using cofinite quantification are strong enough for metatheoretic reasoning, and need not be explicitly strengthened. Strong inversion principles follow (automatically, in Coq) from the induction principles. Although many of the underlying ingredients of our technique have been used before, their combination here yields a significant improvement over other methodologies using first-order representations, leading to developments that are faithful to informal practice, yet require no external tool support and little infrastructure within the proof assistant.We have carried out several large developments in this style using the Coq proof assistant and have made them publicly available. Our developments include type soundness for System F<: and core ML (with references, exceptions, datatypes, recursion, and patterns) and subject reduction for the Calculus of Constructions. Not only do these developments demonstrate the comprehensiveness of our approach; they have also been optimized for clarity and robustness, making them good templates for future extension.

MrSpidey is a user-friendly, i n teractive static debugger for Scheme. A static debugger supplements the standard debugger by analyzing the program and pinpointing those program operations that may cause run-time errors such as dereferencing the null pointer or applying non-functions. The program analysis of MrSpidey computes value set descriptions for each term in the program and constructs a value ow graph connecting the set descriptions. Using the set descriptions, MrSpidey can identify and highlight potentially erroneous program operations, whose cause the programmer can then explore by selectively exposing portions of the value ow graph.

This paper presents the design of Zombie, a dependently-typed programming language that uses an adaptation of a congruence closure algorithm for proof and type inference. This algorithm allows the type checker to automatically use equality assumptions from the context when reasoning about equality. Most dependently-typed languages automatically use equalities that follow from beta-reduction during type checking; however, such reasoning is incompatible with congruence closure. In contrast, Zombie does not use automatic beta-reduction because types may contain potentially diverging terms. Therefore Zombie provides a unique opportunity to explore an alternative definition of equivalence in dependently-typed language design. Our work includes the specification of the language via a bidirectional type system, which works "up-to-congruence,'' and an algorithm for elaborating expressions in this language to an explicitly typed core language. We prove that our elaboration algorithm is complete with respect to the source type system, and always produces well typed terms in the core language. This algorithm has been implemented in the Zombie language, which includes general recursion, irrelevant arguments, heterogeneous equality and datatypes.

Most dependently-typed programming languages either require that all expressions terminate (e.g. Coq, Agda, and Epigram), or allow infinite loops but are inconsistent when viewed as logics (e.g. Haskell, ATS, Ωmega. Here, we combine these two approaches into a single dependently-typed core language. The language is composed of two fragments that share a common syntax and overlapping semantics: a logic that guarantees total correctness, and a call-by-value programming language that guarantees type safety but not termination. The two fragments may interact: logical expressions may be used as programs; the logic may soundly reason about potentially nonterminating programs; programs can require logical proofs as arguments; and "mobile" program values, including proofs computed at runtime, may be used as evidence by the logic. This language allows programmers to work with total and partial functions uniformly, providing a smooth path from functional programming to dependently-typed programming.