Lambda the Ultimate

Towards a Mechanized Metatheory of Standard ML, Daniel K. Lee, Karl Crary, and Robert Harper.

We present an internal language with equivalent expressive power to Standard ML, and discuss its formalization in LF and the machine-checked verification of its type safety in Twelf. The internal language is intended to serve as the target of elaboration in an elaborative semantics for Standard ML in the style of Harper and Stone. Therefore, it includes all the programming mechanisms necessary to implement Standard ML, including translucent modules, abstraction, polymorphism, higher kinds, references, exceptions, recursive types, and recursive functions. Our successful formalization of the proof involved a careful interplay between the precise formulations of the various mechanisms, and required the invention of new representation and proof techniques of general interest.

The way that most programming languages end up working is by defining a smaller core language to which the constructions in the base language are translated. This means that you can prove the type-safety of a programming language by showing that the internal language is type safe, and that every type-correct program in the full language translates to a type-correct expression in the internal language. In this paper, the authors carried out the mechanization of a core language for SML. The next step is mechanizing the correctness of an elaborator from SML to this core language, and then full, no-foolin' SML will have a fully machine-checked correctness proof.