LeanCSP: A Framework for Certifying Constraint Reformulation and Solving in Lean
Authors: Pablo Manrique, Stefan Szeider
Organizations: Algorithms and Complexity Group TU Wien, Vienna, Austria
Abstract
Constraint programming is a core technology for solving complex combinatorial problems in scheduling, planning, configuration, and verification. Trusting its results therefore demands guarantees at two levels: that reformulations applied beforehand are semantics-preserving, and that solvers produce correct answers. In this work, we introduce a framework that addresses both verification levels in the Lean theorem prover: it can be used to prove formulation-level properties, such as equivalence, equisatisfiability, and the correctness of symmetry-breaking constraints, parametrically for entire problem families; and to check solver-produced certificates for individual instances via translation backends to external formats such as MiniZinc, SMT-LIB, and OPB. Combining both levels yields an end-to-end workflow that establishes the satisfiability or unsatisfiability of a constraint problem without trusting the external solver. Experimental results show that our framework's verified symmetry breaking also pays off in practice: a single parametric proof per problem family, reused across all instance sizes, reduces solver search effort by a factor of up to 2x10^7, while the entire in-Lean certification stays affordable, taking at most a few minutes for our largest instances.
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.
MerLean-Prover is an end-to-end Lean4 theorem prover that replaces sorry declarations with kernel-checkable proofs. It is built from three agent types (Planning, Check, and Lean) composed by a recursive outer loop whose unit of revision is the proof plan itself, and uses no fine-tuning, no custom RL objective, and no theorem-specific scaffolding. On FormalQualBench, a benchmark of 23 PhD-qualifying-exam theorems, MerLean-Prover solves 10/23, surpassing the strongest published open-source baseline (OpenGauss, 8/23). On Putnam2025, the same harness closes 12/12 with substantially lower total wall-clock than the next-best system that closes the full set. The harness also transfers to smaller models: Sonnet closes all four tested FormalQualBench problems, and Haiku closes the two short ones. These results suggest that harness design is a central factor in end-to-end Lean4 theorem proving, alongside raw model capability, and that a relatively simple harness can already be effective.
Large Language Models (LLMs) have shown remarkable promise in translating and reformulating complex mathematical optimization problems across modeling languages. However, validating such transformations through empirical solver executions alone is unreliable, as solver outcomes may be affected by local minima, structural timeouts, numerical artifacts, and subtle semantic divergence between formulations. We introduce SOVER, an LLM-assisted SMT framework that separates semantic mapping from formal certification: Z3 checks domain cross-feasibility and global objective-order preservation for mixed-integer linear formulations, while dReal provides tolerance-aware feasibility/range and ε-argmin checks for continuous nonlinear formulations. We also introduce NLEquiv-150, a public benchmark of 100 equivalent and 50 deliberately hard non-equivalent nonlinear reformulation pairs. With LLM-extracted mappings, SOVER classifies 149/150 pairs (99.33%) correctly, including all 50 hard negatives; the sole error is an incomplete mapping extraction.