General Automation in Coq through Modular Transformations

TL;DR

This paper introduces a modular methodology to transform Coq proof goals into first-order logic, enabling the use of automatic theorem provers to solve complex goals more easily.

Contribution

It presents a novel modular approach for transforming Coq goals into first-order logic, facilitating automation in a proof assistant based on Type Theory.

Findings

Transformations extend local context with first-order assertions.

Enabled automatic solving of non-trivial Coq goals.

Framework allows combining multiple transformations for better automation.

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.