GMeta: A Generic Formal Metatheory Framework for First-Order Representations

Abstract

Abstract. This paper presents GMeta: a generic framework for firstorder representations of variable binding that provides once and for all many of the so-called infrastructure lemmas and definitions required in mechanizations of formal metatheory. The key idea is to employ datatypegeneric programming (DGP) and modular programming techniques to deal with the infrastructure overhead. Using a generic universe for representing a large family of object languages we define datatype-generic libraries of infrastructure for first-order representations such as locally nameless or de Bruijn indices. Modules are used to provide templates: a convenient interface between the datatype-generic libraries and the endusers of GMeta. We conducted case studies based on the POPLmark challenge, and showed that dealing with challenging binding constructs, like the ones found in System F<:, is possible with GMeta. All of GMeta's generic infrastructure is implemented in the Coq theorem prover. Furthermore, due to GMeta's modular design, the libraries can be easily used, extended, and customized by users.

Generic datatypes à la carte

“…Licata and Harper [16] and Keuchel and Jeuring [13] define universes for datatypes with binding in Agda. Lee et al [15] develop a framework for first-order representations of variable binding in Coq that is based on the universe of regular tree types [23] and that provides many of the socalled infrastructure lemmas required when mechanizing programming language meta-theory.…”

“…The GMeta [15] framework includes a standalone tool that also performs this derivation for Coq. Similarly we also like to be able to derive instances for the Container and Polynomial classes automatically.…”

Two Recent Approaches to Mechanical Developments of Formal Metatheory

“…To deal with the combinatorial explosion of infrastructure operations and lemmas, Lee et al [8] proposed a generic framework called GMeta for first-order representations of variable binding. GMeta makes use of datatype-generic programming (DGP) and modular programming techniques.…”

Generic conversions of abstract syntax representations

“…Oliveira et al [7] are working on a datatype-generic framework GMeta for the mechanization of formal meta-theory of first-order representations. They make use of a universe construction to represent abstract syntax types and give different first-order interpretations.…”

“…Datatype-generic treatment of abstract syntax has already been addressed in the literature [4,7,8,12] using different approaches to abstract syntax with different trade-offs in terms of elegance, usability and expressivity.…”