FORALL-LEAN-AGENT for Auditable Reasoning in Formal Mathematics and Software Verification

Large language models have achieved striking results in interactive theorem proving, particularly in Lean. However, most benchmarks for LLM-based proof automation are drawn from mathematics in the Mathlib ecosystem, whereas proofs in software verification are developed inside definition-rich codebases with substantial project-specific libraries. We introduce VeriSoftBench, a benchmark of 500 Lean 4 proof obligations drawn from open-source formal-methods developments and packaged to preserve realistic repository context and cross-file dependencies. Our evaluation of frontier LLMs and specialized provers yields three observations. First, provers tuned for Mathlib-style mathematics transfer poorly to this repository-centric setting. Second, success is strongly correlated with transitive repository dependence: tasks whose proofs draw on large, multi-hop dependency closures are less likely to be solved. Third, providing curated context restricted to a proof's dependency closure improves performance relative to exposing the full repository, but nevertheless leaves substantial room for improvement. Our benchmark and evaluation suite are released at https://github.com/utopia-group/VeriSoftBench.

Proof auto-formalization translates natural-language (NL) theorems and proofs into a formal language (FL) such as Lean, enabling mechanical verification. Despite rapid progress, research-level proofs often depend on concepts missing from leading proof assistant libraries (e.g., Lean's Mathlib), and successful compilation does not guarantee that a translation preserves the theorem's meaning or the proof's reasoning. Furthermore, aligned NL-FL training data are scarce, and leading agents often rely on costly frontier models and manually engineered harnesses. To address the above issues, we present AIProver, an agentic framework for autonomous proof auto-formalization and proof synthesis (AFPS) that jointly post-trains a 119B open-weight language model and evolves its agentic, tool-calling harness with HarnessEvolve. Verifiers assess type correctness, proof completeness, and semantic correctness, returning rewards and diagnostic certificates that drive model fine-tuning and alternating reinforcement learning via symbolic feedback and HarnessEvolve, a certificate-driven evolutionary search over the whole harness control flow that re-tailors the harness to the updated model. For research-level training and evaluation, we introduce LoCoBench, 58.9k instances from Mathlib, CSLib, Mizar Math Library, and a bounded-arithmetic textbook, with a 771-instance validation split whose theorem-proof pairs have no public Lean formalization. Against 39 frameworks spanning AFPS agents, frontier LLMs, and coding agents, AIProver lifts pass@4 semantic correctness over its Leanstral-1.5 base from 15.7% to 36.7% and outperforms every other open-weight system and Aristotle. As a Claude Code and Codex skill, it lifts their semantic correctness from 41.9% and 34.1% to 79.8% and 62.4%, respectively. Further, it is also 24% cheaper than Numina-Lean-Agent, pushing the accuracy-cost frontier of research-level AFPS.

Designing effective Lean proof agents is a central challenge in formal mathematical reasoning. Beyond building stronger provers, recent work emphasizes the workflow around Lean: how an agent decomposes proof obligations, uses tools and compiler feedback, diagnoses failures, repairs proofs, and maintains structured proof context. Motivated by code-level self-evolving agents, we study whether such workflows can be evolved rather than hand-designed. We present a self-evolving Lean proof agent in which a small fixed, trusted runtime wraps a fully mutable workspace: the proof workflow, prompts, and tools. Unlike most self-evolving systems, which optimize against a fixed external benchmark, our system coevolves the agent and its benchmark. Between generations, the highest-scoring agent (the champion) revises the active task distribution through a mastery-throttled curriculum update that introduces harder proof obligations only after the current level is mastered, and a single-anchor recalibration re-runs the champion on the updated benchmark to keep scores comparable as difficulty rises. All evolution stays inside a Lean-grounded verification loop: however the agent rewrites itself, a success counts only when its behavior yields Lean-verified proofs under a trusted snapshot, and each attempt must emit a machine-readable, Lean-grounded proof context whose representation may evolve but whose groundedness is enforced. We run the coevolving trajectory and a fixed-benchmark baseline for 15 active generations and compare them on a held-out miniF2F test split. The best coevolving agent reaches a 45.1% held-out solve rate, versus 12.7% for the seed and 32.0% for the best fixed-benchmark agent, showing that verifier-grounded self-evolution can improve Lean proof workflows under a coevolving benchmark.