Improving Coq Propositional Reasoning Using a Lazy CNF Conversion Scheme

Abstract

Abstract. In an attempt to improve automation capabilities in the Coq proof assistant, we develop a tactic for the propositional fragment based on the DPLL procedure. Although formulas naturally arising in interactive proofs do not require a state-of-the-art SAT solver, the conversion to clausal form required by DPLL strongly damages the performance of the procedure. In this paper, we present a reflexive DPLL algorithm formalized in Coq which outperforms the existing tactics. It is tightly coupled with a lazy CNF conversion scheme which, unlike Tseitin-style approaches, does not disrupt the procedure. This conversion relies on a lazy mechanism which requires slight adaptations of the original DPLL. As far as we know, this is the first formal proof of this mechanism and its Coq implementation raises interesting challenges.

QED at Large: A Survey of Engineering of Formally Verified Software

“…The tool Hipster (Johansson et al, 2014;Valbuena and Johansson, 2015;Johansson, 2017) for Isabelle/HOL integrates theory exploration directly with an ITP. Other examples of useful general-purpose automation include simple general-purpose proof automation (Coq Development Team, 1999Team, -2018bZhan, 2016;Lindblad and Benke, 2006), rewriting (Coq Development Team, 1999Team, -2018bNipkow, 1989), and solving logical fragments (Paulson, 1999;Lescuyer and Conchon, 2009;Hurd, 2003;Kumar et al, 1991;Busch, 1994;Dahn et al, 1997;Hurd, 1999), and techniques for reasoning about executable specficiations (Barthe and Courtieu, 2002), as well as an implementation of a generalization of congruence closure to dependent type theory (Selsam and de Moura, 2017). In addition, Chapter 6 describes general-purpose automation and tooling for proof reuse (Section 6.4.3), as well as general-purpose automation built on type classes and canonical structures (Section 6.2.1).…”

A Modular Integration of SAT/SMT Solvers to Coq through Proof Witnesses

“…This is the approach followed in [10]. It has the advantage to validate the algorithms at work in the prover but is sensitive to any change, any optimization in the proof search.…”

“…In this case, this means implementing a whole, sufficiently efficient, SAT/SMT solver as a Coq function, and then proving it correct. This approach is followed in the Ergo-Coq effort [10].…”

SAT Solving for Termination Proofs with Recursive Path Orders and Dependency Pairs

“…For some recent examples see: Jefferson in [28] to model sophisticated propagators for constraint programming problems, Feydy et al in [16] to model difference constraints and to design finite domain propagators, Lescuyer and Conchon in [37] to provide proofs by reflection in the Coq theorem prover, Gotlieb in [25] to model a verification problem (where it is illustrated that constraint programming can compete with other techniques based on SAT and SMT solvers), Bofill et al in [7] to model Max-SAT problems and to encode them as pseudo-Boolean constraints, and there are many more.…”