No abstract

Tait's method (a.k.a. proof by logical relations) is a powerful proof technique frequently used for showing foundational properties of languages based on typed λ-calculi. Historically, these proofs have been extremely difficult to formalize in proof assistants with weak meta-logics, such as Twelf, and yet they are often straightforward in proof assistants with stronger meta-logics. In this paper, we propose structural logical relations as a technique for conducting these proofs in systems with limited meta-logical strength by explicitly representing and reasoning about an auxiliary logic. In support of our claims, we give a Twelf-checked proof of the completeness of an algorithm for checking equality of simply typed λ-terms.

Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of Martin-Löf's intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel's T follows indirectly; both have been formalized in a prototypical extension of Twelf.