Mechanizing metatheory in a logical framework

Abstract

The LF logical framework codifies a methodology for representing deductive systems, such as programming languages and logics, within a dependently typed λ-calculus. In this methodology, the syntactic and deductive apparatus of a system is encoded as the canonical forms of associated LF types; an encoding is correct (adequate) if and only if it defines a compositional bijection between the apparatus of the deductive system and the associated canonical forms. Given an adequate encoding, one may establish metatheoretic properties of a deductive system by reasoning about the associated LF representation. The Twelf implementation of the LF logical framework is a convenient and powerful tool for putting this methodology into practice. Twelf supports both the representation of a deductive system and the mechanical verification of proofs of metatheorems about it.The purpose of this paper is to provide an up-to-date overview of the LF λ-calculus, the LF methodology for adequate representation, and the Twelf methodology for mechanizing metatheory. We begin by defining a variant of the original LF language, called Canonical LF, in which only canonical forms (long βη-normal forms) are permitted. This variant is parameterized by a subordination relation, which enables modular reasoning about LF representations. We then give an adequate representation of a simply typed λ-calculus in Canonical LF, both to illustrate adequacy and to serve as an object of analysis. Using this representation, we formalize and verify the proofs of some metatheoretic results, including preservation, determinacy, and strengthening. Each example illustrates a significant aspect of using LF and Twelf for formalized metatheory.

Reasoning with Higher-Order Abstract Syntax and Contexts: A Comparison

“…Since the schema eCtx in fact contains at least as much information as the schema eqCtx, we should be allowed to pass a context of schema eCtx when a context of schema eqCtx is required. This is achieved by incorporating context subsumption in Beluga (see [Sch00,HL07] for an introduction to context subsumption).…”

“…Weakening and strengthening is handled automatically. For a more detailed explanation regarding the formalization of proofs in the Twelf system and context subsumption, we refer the reader to [HL07].…”

Hereditary substitutions for simple types, formalized

“…Their property to preserve canonical forms during substitution makes them a nice approach to an implementation of substitutions for Logical Frameworks [9,12] and Higher-Order Abstract Syntax [15]. Abel [3] already noticed that the fact that hereditary substitutions are structurally recursive makes it easy to automatically check the termination of the algorithm they provide.…”

Using Vampire in Soundness Proofs of Type Systems

“…Despite several existing solutions to the PoplMark challenge [10,3,21,4], there has been no fully automated solution to date. The probably highest potential of full automation among the set of solutions submitted to the PoplMark challenge is the Twelf approach [6]. Twelf is a specialpurpose theorem prover for properties of logics and programming languages.…”

“…Sledgehammer is a tool within the general-purpose theorem prover Isabelle [12] for automatically solving individual proof goals within larger proofs 6 . Sledgehammer translates proof goals to first order logic and sends them to Vampire and to other several automated theorem provers and SMT solvers, accompanying them with heuristically selected relevant facts and lemmas from the original Isabelle theory.…”