Scalable Fine-Grained Proofs for Formula Processing

Abstract

We present a framework for processing formulas in automatic theorem provers, with generation of detailed proofs. The main components are a generic contextual recursion algorithm and an extensible set of inference rules. Clausification, skolemization, theoryspecific simplifications, and expansion of 'let' expressions are instances of this framework. With suitable data structures, proof generation adds only a linear-time overhead, and proofs can be checked in linear time. We implemented the approach in the SMT solver veriT. This allowed us to dramatically simplify the code base while increasing the number of problems for which detailed proofs can be produced, which is important for independent checking and reconstruction in proof assistants. To validate the framework, we implemented proof reconstruction in Isabelle/HOL. 1 Introduction An increasing number of automatic theorem provers can generate certificates, or proofs, that justify the formulas they derive. These proofs can be checked by other programs and shared across reasoning systems. It might also happen that a human user would want to inspect

Alethe: Towards a Generic SMT Proof Format (extended abstract)

“…It was proposed as an easy-to-produce format with a term language very close to SMT-LIB [5], the standard input language of SMT solvers, and rules with a varying level of granularity, allowing implicit proof steps in the proof and thus relying on powerful proof checkers capable of filling the gaps. Since then the format has been refined and extended [2]. It is now mature, supports coarse-and fine-grained proof steps capturing SMT solving for the SMT-LIB logic UFLIRA 2 and can be reconstructed by the proof assistants Coq [1,11] and Isabelle [12,14].…”

“…Since then the format has been refined and extended [2]. It is now mature, supports coarse-and fine-grained proof steps capturing SMT solving for the SMT-LIB logic UFLIRA 2 and can be reconstructed by the proof assistants Coq [1,11] and Isabelle [12,14]. In particular, the integration with Coq was also used as a bridge for the reconstruction of proofs from the SMT solver CVC4 [3] in Coq, where its proofs in the LFSC format [15] were first translated into the veriT format before reconstruction.…”

Conflict-Driven Satisfiability for Theory Combination: Lemmas, Modules, and Proofs

“…Corollary 1 For all signatures Σ = (S, F) and assignments J , if either (1) for all terms t ∈ G(J ) there is an assignment (t←c) ∈ J , or (2) for all distinct terms t, u ∈ G s (J ) of sort s ∈ S \ {prop} there is an assignment ((t u)←b) ∈ J , then fv Σ (G(J )) = fv Σ (J ).…”

“…The DPLL(T ) or CDCL(T ) paradigm naturally supports the generation of proofs by resolution, where the theory lemmas are plugged in as leaves with black-box subproofs [3,12,27,38]. This style has been implemented in solvers such as Z3 [3], veriT [2,27], and CVC4 [38] and extended in several ways (e.g., [2,38]). In CDSAT, the CDCL-based SAT solver loses its centrality as the only conflict-driven component, and all theory modules contribute directly to the proof, including new terms.…”

A Flexible Proof Format for SAT Solver-Elaborator Communication

“…Notable examples outside SAT solving include the LFSC format for SMT solving [23] and the TSTP format for classical first-order ATPs [24]. In particular, the recent work on the veriT SMT solver [1] is motivated by similar rationales as that for the FRAT toolchain; the key insight is that a proof production pipeline is often easier to optimize on the solver side than on the elaborator side, as the former has direct access to many types of useful information.…”