LLM2Cert: LLM Synthesis of Certificates with SMT/Lean Verification

Lyapunov functions and barrier certificates give state-space guarantees of stability and safety, but synthesizing and verifying them---especially for nonlinear, high-dimensional systems---remains an open challenge. Existing pipelines verify candidates with numerical solvers (sum-of-squares programs), interval methods, or satisfiability-modulo-theories (SMT) solvers, which often require relaxations, are conservative, or scale poorly with model dimension and nonlinearity. These limitations have also shaped benchmark design: standard test suites are dominated by low-dimensional polynomial systems chosen to suit what solvers can handle rather than what engineering applications require. We address both sides of this challenge. First, we introduce DynaCertBench, a benchmark of 105 instances: 80 non-redundant systems curated from the literature, extended with high-dimensional and control-constrained variants and new nonlinearly coupled systems, comprehensively spanning stability, region-of-attraction, forward-invariance, and safety tasks. Second, we present LLM2Cert, an LLM-driven framework that synthesizes candidate certificates and verifies them against their proof obligations: each obligation is routed to an SMT solver or directly to the Lean Proof-Repair agent. On DynaCertBench, LLM2Cert verifies 81 of 105 instances (77.1%), a 17.1 percentage point improvement over one-shot proposal without feedback. On the 81 fixed certificates, LLM2Cert verifies 22 that All-SMT leaves unresolved.