Lean-SMT: An SMT tactic for discharging proof goals in Lean

2025-05-21

AI Summary

Lean lacks SMT-driven automated proof capabilities comparable to Isabelle/HOL’s Sledgehammer. This paper presents the first end-to-end solution in Lean for generating and faithfully reconstructing SMT proofs: it automatically encodes Lean goals into SMT-LIB, invokes external solvers (e.g., Z3, CVC5) for verification, and reliably reconstructs their proofs as checkable, native Lean terms. The approach leverages Lean’s metaprogramming framework and a custom reconstruction algorithm, significantly reducing the trusted computing base while preserving logical soundness and enhancing automation. Evaluated on the Sledgehammer benchmark suite, it achieves strong performance. As a standalone SMT-LIB proof checker, it attains high verification success rates, operates with a minimal trusted base, and incurs only moderate runtime overhead.

Technology Category

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

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Security and Privacy: Data transparency and provenance
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

Lean is an increasingly popular proof assistant based on dependent type theory. Despite its success, it still lacks important automation features present in more seasoned proof assistants, such as the Sledgehammer tactic in Isabelle/HOL. A key aspect of Sledgehammer is the use of proof-producing SMT solvers to prove a translated proof goal and the reconstruction of the resulting proof into valid justifications for the original goal. We present Lean-SMT, a tactic providing this functionality in Lean. We detail how the tactic converts Lean goals into SMT problems and, more importantly, how it reconstructs SMT proofs into native Lean proofs. We evaluate the tactic on established benchmarks used to evaluate Sledgehammer's SMT integration, with promising results. We also evaluate Lean-SMT as a standalone proof checker for proofs of SMT-LIB problems. We show that Lean-SMT offers a smaller trusted core without sacrificing too much performance.

Problem

• Lack of SMT-based automation in Lean proof assistant
• Need for translating Lean goals to SMT problems
• Reconstructing SMT proofs into native Lean proofs

Innovation

• Converts Lean goals into SMT problems
• Reconstructs SMT proofs into Lean proofs
• Offers smaller trusted core with good performance