Inference-Time Diversity in RL-Trained Lean Theorem Provers: A Diagnostic Study

Modern Lean theorem provers achieve strong performance only with substantial training and inference compute, driven in part by scarce verified proof data and the long reasoning traces of formal proof search, making both supervised fine-tuning (SFT) and sampling expensive. We introduce Pythagoras-Prover, a compute-efficient open-source family of Lean theorem provers built for practical compute budgets. The family spans two generation paradigms: autoregressive models at 4B and 32B parameters, and a first proof-of-concept diffusion-based prover (4B) that iteratively refines Lean proofs at inference time. For training efficiency, we build a Lean-verified corpus stratified into easy, medium, and hard problems for curriculum SFT, so models acquire proof skills progressively from shorter, simpler proofs to longer, harder ones. During SFT, a dynamic proof-reasoning filtering scheme preserves informative proof traces while keeping each instance within an 8k-token context budget. We also introduce Augmented Lean Formalisation (ALF), which expands scarce verified corpora into variants of formal statements, populated via self-distillation for extra training signal without formally verifying every mutated instance. By perturbing known problems while preserving their formal character, ALF reduces reliance on any statement's surface form. Empirically, Pythagoras-Prover-4B surpasses DeepSeek-Prover-V2-671B at pass@32 on MiniF2F-Test (86.1% vs 82.4%) with ~167x fewer parameters, while Pythagoras-Prover-32B sets the open-source state of the art at 93.0% on MiniF2F-Test and solves 93 of 672 PutnamBench problems. We release MiniF2F-ALF, an ALF-mutated contamination-sensitive benchmark on which every evaluated model loses accuracy; here our 32B remains strongest and our 4B matches the prior state of the art, Goedel-Prover-V2-32B.

LLMs have recently achieved strong results on formal proving benchmarks. However, existing evaluations remain heavily concentrated on competition-style problems and often fail to capture how models behave on longer, more dependency-rich mathematical developments. We introduce TheoremBench, a Lean4 benchmark designed to evaluate theorem provers beyond contest settings. The benchmark is built from nearly one hundred classical theorems and is released in two complementary forms: a plain main version containing one target theorem per instance, and a premised version that expands each theorem into a structured family of related proving tasks consisting of the main theorem together with automatically extracted supporting subtheorems. This design enables evaluation of not only whether the final theorem was proved from scratch, but also of partial progress through the internal proof structure of a theorem. Our experiments show that explicit premises substantially improve performance for Lean4-capable prover models. To provide a comprehensive evaluation, we introduce theorem-level coverage and token-efficiency metrics that expose qualitative differences in proof behavior. The results show that current provers remain strongly biased toward easy subtheorems and often solve theorems through long and inefficient tactic traces rather than compact proof plans. TheoremBench therefore provides a more fine-grained view of formal reasoning ability and highlights the importance of structural benchmark design for evaluating Lean4 theorem provers.

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.