[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82556-en":3,"doc-seo-82556-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82556,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","LRAT-Catcher: Importing SAT Solver Certificates into Lean 4 by Reflection","SAT solvers address combinatorial problems beyond the scope of interactive theorem provers by emitting LRAT certificates for independent validation. LRAT-Catcher is a standalone tool that imports a DIMACS instance plus an LRAT certificate into Lean 4 as a reusable theorem. It runs Lean-core’s formally verified LRAT checker via reflection as compiled native code, reducing memory blowups seen in explicit proof-term imports. The tool also composes cube-and-conquer runs into a single unsatisfiability theorem using a cover-completeness LRAT proof, with verified encodings linking CNF results to target combinatorial statements like Schur and Ramsey numbers.","arXiv :2607 .008 15v 1 [ cs .LO] 1 Jul 2026  \nLRAT-Catcher: Importing SAT Solver Certificates into  \nLean 4 by Reflection  \nStefan Szeider  \nAlgorithms and Complexity Group, TU Wien, Vienna, Austria  \n[sz@ac.tuwien.ac.at](sz@ac.tuwien.ac.at)  \nAbstract  \nSAT 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.  \nKeywords. SAT certificates, LRAT, Lean 4, proof by reflection, cube-and-conquer, verified encodings.  \n1 Introduction  \nBoolean satisfiability (SAT) solvers settle combinatorial questions that lie beyond the reach of interactive theorem provers. One such question is the value of the fifth Schur number [5] . When a solver reports a formula unsatisfiable, it emits a certificate in the DRAT format or its checker-friendly refinement LRAT [2] . An independent checker confirms the result without repeating the search. Such a verdict is convincing, but not a theorem inside a proof assistant. It cannot be named, reused as a lemma, or composed with other formal developments. Closing this gap means importing a certificate as a reusable theorem whose statement pins down exactly which formula was refuted.  \nTwo Lean developments already import LRAT certificates. The lrat_proof command of Mathlib turns a DIMACS formula and an LRAT certificate into a theorem by constructing an explicit proof term [12], but it does not implement the resolution asymmetric tautology (RAT) rule. Because it binds the entire formula into the proof term, its memory cost grows with the formula rather than the certificate. It therefore exhausts memory on instances that solvers routinely solve. Lean’s standard library, on the other hand, provides a formally verified LRAT checker with full RAT support, Std.Tactic.BVDecide.LRAT, developed by Böving et al. for the bit-blasting tactic bv_decide [1] . That checker can only be reached through bv_decide on bitvector goals and is not exposed as a standalone certificate import. Its checking performance has not been measured against external checkers.  \nA second line of work proves combinatorial theorems with a SAT solver in the loop. The Empty Hexagon development of Subercaseaux et al. [9] verifies the encoding in Lean but checks the certificate externally with cake  lpr [11] and asserts the resulting unsatisfiability as an axiom after a solve-and-check computation of roughly 3 CPU years. Kirchweger et al. [6] integrate SAT modulo symmetries with Lean for end-to-end verified graph generation, formalizing the symmetry-breaking reasoning inside Lean to certify graph non-existence. The end-to-end formal verification of Keller’s conjecture by Gallicchio et al. [3] comes closest to what we aim for. It pairs verified encodings with verified LRAT and substitutionredundancy checkers. It runs the entire pipeline inside Lean by reflection, evaluating the verified checkers directly as Lean computations rather than emitting a separate proof term. Its encodings are built especially for that proof, and the method is not offered as a reusable certificate-import bridge.  \nOutside Lean, the verified checkers cake  lpr and Lammich’","cbCaijZ6P1G21xKI","https://ap.wps.com/l/cbCaijZ6P1G21xKI","pdf",637351,4,1,11,"English","en",105,"# Introduction\n## Background on SAT certificates and LRAT\n## Related Lean and external verification approaches\n## Contribution and design of LRAT-Catcher\n# Evaluation and results\n## Reflection modes and scalability\n## Schur and Ramsey encodings in Lean","[{\"question\":\"What problem does LRAT-Catcher solve when working with SAT solvers and Lean?\",\"answer\":\"It turns SAT solver LRAT certificates into reusable theorems inside Lean 4, so results can be named, reused as lemmas, and composed with other formal developments rather than remaining external solver verdicts.\"},{\"question\":\"How does LRAT-Catcher check LRAT certificates inside Lean 4?\",\"answer\":\"It imports a DIMACS formula and an LRAT certificate, then executes Lean-core’s formally verified LRAT checker via reflection by compiling it to native code, with options including a kernel-only checking mode.\"},{\"question\":\"How does LRAT-Catcher handle cube-and-conquer workflows?\",\"answer\":\"It runs cube-and-conquer entirely inside Lean, combines per-cube refutations with a cover-completeness certificate, and produces a single unsatisfiability theorem where the cover-completeness certificate is itself an LRAT 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problem does LRAT-Catcher solve when working with SAT solvers and Lean?","Question",{"text":75,"@type":76},"It turns SAT solver LRAT certificates into reusable theorems inside Lean 4, so results can be named, reused as lemmas, and composed with other formal developments rather than remaining external solver verdicts.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does LRAT-Catcher check LRAT certificates inside Lean 4?",{"text":80,"@type":76},"It imports a DIMACS formula and an LRAT certificate, then executes Lean-core’s formally verified LRAT checker via reflection by compiling it to native code, with options including a kernel-only checking mode.",{"name":82,"@type":73,"acceptedAnswer":83},"How does LRAT-Catcher handle cube-and-conquer workflows?",{"text":84,"@type":76},"It runs cube-and-conquer entirely inside Lean, combines per-cube refutations with a cover-completeness certificate, and produces a single unsatisfiability theorem where the cover-completeness certificate 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