PBLean: Pseudo-Boolean Proof Certificates for Lean 4

2026-02-09

AI Summary

This work addresses the challenge of reliably importing VeriPB proof certificates generated by pseudo-Boolean (PB) solvers into Lean 4 and achieving end-to-end trustworthy verification from solver outputs back to the semantics of the original combinatorial problem. To this end, we present the first formalization in Lean 4 of a reflective proof checker that fully supports the VeriPB kernel rules. By integrating native code compilation, verified encoding transformations, and cutting-plane derivation techniques, our approach efficiently handles large-scale proofs comprising tens of thousands of inference steps while avoiding the memory bottlenecks associated with explicit proof-term construction. This method bridges the trust gap between solver output and formal semantics, producing reusable and composable Lean theorems, and demonstrates both effectiveness and scalability across multiple combinatorial problems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity

Abstract

We present PBLean, a method for importing VeriPB pseudo-Boolean (PB) proof certificates into Lean 4. Key to our approach is reflection: a Boolean checker function whose soundness is fully proved in Lean and executed as compiled native code. Our method scales to proofs with tens of thousands of steps that would exhaust memory under explicit proof-term construction. Our checker supports all VeriPB kernel rules, including cutting-plane derivations and proof-by-contradiction subproofs. In contrast to external verified checkers that produce verdicts, our integration yields Lean theorems that can serve as composable lemmas in larger formal developments. To derive theorems about the original combinatorial problems rather than about PB constraints alone, we support verified encodings. This closes the trust gap between solver output and problem semantics since the constraint translation and its correctness proof are both formalized in Lean. We demonstrate the approach on various combinatorial problems.

Problem

• Pseudo-Boolean proofs
• proof certificates
• Lean 4
• formal verification
• combinatorial problems

Innovation

• reflection
• Pseudo-Boolean proofs
• proof certificates
• formal verification
• Lean 4