Mechanizing Metatheory Without Typing Contexts

Abstract

When mechanizing the metatheory of a programming language, one usually needs many lemmas proving structural properties of typing judgments, such as permutation and weakening. Such structural lemmas are sometimes unnecessary if we eliminate typing contexts by expanding typing judgments into their original hypothetical proofs. This technique of eliminating typing contexts, which has been around since Church [4], is based on the view that entailment relations, such as typing judgments, are just syntactic tools for displaying only the hypotheses and conclusion of a hypothetical proof while hiding its internal structure.In this paper, we apply this technique to the POPLmark challenge [1] and experimentally evaluate its efficiency by formalizing System F <: in the Coq proof assistant in a number of different ways. An analysis of our Coq developments shows that eliminating typing contexts produces a more significant reduction in both the number of lemmas and the count of tactics than the cofinite quantification, one of the most effective ways of simplifying the mechanization involving binders. Our experiment with System F <: suggests three guidelines to follow when applying the technique of eliminating typing contexts.

Polymorphic System I

“…Notice, however, that since the system is given in Church-style (i.e. variables have their type written), the context is redundant [19,23]. Hence, we may write "t has type A" with no ambiguity.…”

Extensional proofs in a propositional logic modulo isomorphisms

“…We use a presentation of typing rules without explicit context following [20,24], hence the typing judgments have the form r : A. The well-typed preterms are called terms.…”

The probability of non-confluent systems

“…A term r is typable if there exists a type A such that r : A. Following [19,22], we use a presentation of typed lambda-calculus without contexts and where each variable occurrence is labelled by its type, such as. λ x A .x A or λ x A .y B .…”