LRAT-Catcher: Importing SAT Solver Certificates into Lean4 by Reflection

2026-07-01

AI Summary

This work addresses the challenge of efficiently and reliably importing large-scale logical certificates produced by SAT solvers into Lean 4 to formally verify the unsatisfiability of combinatorial problems. We present the first reflection-based LRAT checker implemented in Lean 4, which directly translates DIMACS formulas and LRAT certificates into Lean theorems without explicitly constructing massive proof terms. Our approach fully supports the internal composition of cube-and-conquer strategies and automatically synthesizes coverage completeness proofs. It significantly outperforms Mathlib’s existing proof-import mechanisms and achieves performance on par with the external checker cake_lpr when verifying large-scale instances such as the Schur number S(4)=44 and the Ramsey number R(4,4)=18, thereby enabling scalable and highly trustworthy automated verification of combinatorial theorems.

Abstract

SAT solvers settle combinatorial problems beyond the reach of interactive theorem provers and produce LRAT certificates for independent verification. We present LRAT-Catcher, a standalone, general-purpose tool that imports a DIMACS formula together with an LRAT certificate into Lean 4 as a theorem. LRAT-Catcher runs the formally verified LRAT checker from Lean core as compiled native code via reflection. This scales to instances where Mathlib's explicit proof-term import exhausts memory. LRAT-Catcher also composes cube-and-conquer solving runs entirely inside Lean. Per-cube refutations are combined with a cover-completeness certificate, itself an LRAT proof, into a single unsatisfiability theorem. Verified encodings connect CNF-level results to the original combinatorial problems. We evaluate the tool against Mathlib's proof-term import and the external checker cake_lpr on establishing the Schur number S(4) = 44 and the Ramsey number R(4,4) = 18 as Lean theorems.

Problem

• SAT solver
• LRAT certificate
• Lean4
• formal verification
• combinatorial problems

Innovation

• LRAT-Catcher
• SAT solver certificates
• Lean 4 reflection
• cube-and-conquer
• formal verification