General Automation in Coq through Modular Transformations

Abstract

Whereas proof assistants based on Higher-Order Logic benefit from external solvers' automation, those based on Type Theory resist automation and thus require more expertise. Indeed, the latter use a more expressive logic which is further away from first-order logic, the logic of most automatic theorem provers. In this article, we develop a methodology to transform a subset of Coq goals into first-order statements that can be automatically discharged by automatic provers. The general idea is to write modular, pairwise independent transformations and combine them. Each of these eliminates a specific aspect of Coq logic towards first-order logic. As a proof of concept, we apply this methodology to a set of simple but crucial transformations which extend the local context with proven first-order assertions that make Coq definitions and algebraic types explicit. They allow users of Coq to solve non-trivial goals automatically. This methodology paves the way towards the definition and combination of more complex transformations, making Coq more accessible. * This work is funded by a Nomadic Labs-Inria collaboration.

Compositional Pre-processing for Automated Reasoning in Dependent Type Theory

“…The present work extends and generalizes a previous work [7] that combined some pre-processing tactics with the SMTCoq automation tool. Notably, pre-processing is now completely untied from the backend, which allows for an association with arbitrary tactics, as opposed to just the SMTCoq plugin.…”

“…We also use the version of the tactic implemented in [7], which is more exhaustive and faster when the context contains a reasonable number of potential instances (two or three) but can be slower when the instances are numerous (exponential in the number of instances). Bobot and Paskevich proposed a complete transformation to encompass polymorphism [8].…”

Modular pre-processing for automated reasoning in dependent type theory

“…The present paper significantly extends a preliminary description by some of the authors [4] with the following novel contributions: we effectively target SMT solvers, for their theory reasoning capabilities, which stresses the need for adequate pre-processing of arithmetic and Boolean connectives (under uninterpreted function symbols). This motivated the design Trakt, one of our main contributions here; we implement other new transformations, such as eliminators for algebraic data-types, and we improve the robustness of the whole strategy; we evaluate the resulting snipe tactic both on performance and expressivity benchmarks.…”

“…The role of the Trakt transformation is to target this fragment, which is the object of § 5.1. More details about the other transformations can be found in previous work [4].…”