FineProofs-RL: Verifier-Centric Theorem Proving

• FineProofs-RL is a family of reinforcement learning methods that leverages symbolic verifiers and formal feedback loops to optimize theorem proving.
• It employs multi-turn interactions, token-level credit assignment, and hierarchical lemma generation to refine proof strategies.
• Empirical results show significant improvements in pass@k metrics across systems like Lean and Isabelle, highlighting its practical impact on formal verification.

FineProofs-RL denotes a class of reinforcement-learning methods for proof tasks in which a symbolic verifier, proof assistant, or closely aligned proof-checking mechanism supplies the training signal. In the cited literature, the term is used both for reinforcement learning explicitly targeted at formal proof generation with a verifier in the loop and for reinforcement learning for theorem proving that exploits fine-grained, process-level proof signals; related work applies the same design logic to hierarchical lemma generation, proof-tree correction, and proof simplification (Ji et al., 11 Jul 2025, Kim et al., 18 Jun 2026, Dong et al., 2024).

1. Conceptual scope

In Lean-based formal theorem proving, FineProofs-RL is instantiated by treating the Lean 4 verifier as the environment, letting the policy emit long reasoning traces that contain Lean code, and feeding verifier success or error diagnostics back into the context for further correction. The central mechanism is therefore not merely post hoc checking, but verifier-integrated interaction over multi-turn trajectories in which proof attempts, compiler feedback, and revisions are all part of the optimization target (Ji et al., 11 Jul 2025).

In the process-verified formulation, the emphasis shifts from sparse outcome rewards to dense, verifier-grounded process supervision. Lean parses a generated proof into tactics, identifies locally sound steps and the earliest failing step, and thereby becomes a symbolic process oracle rather than only a terminal correctness oracle. The resulting reward structure is explicitly rooted in Lean’s elaboration and type-theoretic semantics (Kim et al., 18 Jun 2026).

A third usage appears in Isabelle/HOL, where FineProofs-RL is identified with hierarchical proof decomposition. Here the policy generates conditional proofs that may invoke new lemmas, proves those lemmas as child nodes, and is rewarded not only for complete theorem proofs but also for correct intermediate lemmas, including lemmas not present in the supervised training set (Dong et al., 2024). Taken together, these usages suggest that FineProofs-RL is best understood as a verifier-centered family of RL procedures for formal reasoning, rather than a single algorithm.

2. Reinforcement-learning formulations and reward design

A common feature across FineProofs-RL systems is the explicit formalization of theorem proving as an MDP whose states include structured proof context and whose rewards derive from symbolic verification. In Leanabell-Prover-V2, the state at turn $t$ includes the formal statement, prior chain-of-thought, the latest <code>...</code> block, and Lean feedback inside <interpreter>...</interpreter>. Actions are next-token generations by $\pi_\theta$, transitions are induced by text concatenation plus verifier invocation, and the episodic objective is

$$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$

The policy is optimized with DAPO, a token-level clipped-ratio objective, and a feedback token mask $m_{i,t}$ that removes verifier-injected tokens from SFT and RL losses. The reward is deliberately simple:

$$R_i = R_{\mathrm{format}}(i) + R_{\mathrm{status}}(i),$$

with $R_{\mathrm{format}}(i)=0.2$ if the sample contains at least one syntactically correct <code>...</code> block that triggers the verifier, and $R_{\mathrm{status}}(i)\in\{0,1\}$ depending on compilation success or failure (Ji et al., 11 Jul 2025).

Process-verified FineProofs-RL uses a denser reward geometry. A generated Lean proof $Y$ is parsed into tactics $(T_1,\dots,T_{N(Y)})$, Lean identifies the earliest failing tactic index $j$, and tactic-level scores are assigned by first-error propagation: if $\pi_\theta$0, then every tactic receives score $\pi_\theta$1; if $\pi_\theta$2, then locally sound tactics before $\pi_\theta$3 receive $\pi_\theta$4, while the first error and all subsequent tactics receive $\pi_\theta$5. These process rewards are combined with outcome-level advantages in a GRPO-style objective, and tactic-level credit is applied only to the first token of each tactic:

$\pi_\theta$6

This formulation is designed to preserve causal consistency once a proof prefix becomes invalid (Kim et al., 18 Jun 2026).

In ProD-RL, the RL update is implemented as weighted log-likelihood over conditional proofs rather than token-level PPO-style optimization. A proof node receives weight

$\pi_\theta$7

with $\pi_\theta$8 if the node is not locally correct, and the effective policy objective is

$\pi_\theta$9

The distinctive aspect is that correct lemma nodes are retained and trained even when the parent theorem fails, so intermediate verifier-accepted subresults become RL signal rather than discarded rollouts (Dong et al., 2024).

3. Verifier interaction patterns

The major FineProofs-RL systems differ primarily in the granularity at which they expose proof structure to the verifier and in the form of the resulting credit assignment.

System	Proof granularity	Core verifier interaction
Leanabell-Prover-V2 (Ji et al., 11 Jul 2025)	Long CoT with embedded Lean code	Multi-turn compile/diagnose/revise loop
Process-Verified RL (Kim et al., 18 Jun 2026)	Tactic sequence in Lean AST	Local soundness, earliest failing step, first-error propagation
ProD-RL (Dong et al., 2024)	Conditional proof tree in Isabelle/HOL	Local/global correctness of theorem and child lemmas
ProofNet++ (Ambati, 30 May 2025)	Proof tree and failed subtrees	Step-level binary verifier reward plus self-correction

Leanabell-Prover-V2 exemplifies the “verifier-integrated multi-turn loop.” The model emits a proof candidate inside <code>...</code>, Lean returns either “Compilation Success!” or a structured error log, and the next turn conditions on the accumulated reasoning trace and feedback. This produces trajectories of attempt $$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$0 feedback $$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$1 revision, which are then optimized directly by RL (Ji et al., 11 Jul 2025).

Process-Verified RL instead treats Lean’s elaboration info-tree as the central supervisory object. The proof assistant parses tactics, marks which steps elaborate successfully, and exposes the earliest failing tactic. Because the process reward is attached to tactic boundaries and propagated from the first error onward, the verifier is used not only to score outcomes but also to define a causal error structure over the proof process (Kim et al., 18 Jun 2026).

ProD-RL introduces a tree-structured interaction pattern. The model can emit <invoke>...</invoke> lemma statements inside a conditional proof, after which those lemmas become child proof obligations. Local correctness is checked with proposed lemmas temporarily registered as facts, while global correctness requires assembly of a valid proof tree whose children are themselves globally correct. The method therefore trains the policy to decompose a theorem into reusable subgoals rather than to emit a flat proof string (Dong et al., 2024).

ProofNet++ occupies an intermediate neuro-symbolic position. It combines a transformer LLM, a Symbolic Reasoning Interface, formal verification in Lean 4 or HOL Light, symbolic proof tree supervision, and a Self-Correction module that proposes repairs for failed nodes or subtrees, with each proposed repair validated by the verifier (Ambati, 30 May 2025).

4. Representative systems and empirical performance

Leanabell-Prover-V2 is a 7B prover that posttrains Kimina-Prover-Preview-Distill-7B and DeepSeek-Prover-V2-7B with cold-start SFT followed by verifier-integrated RL. On MiniF2F-test, the Kimina baseline is reported at 63.1% pass@32 and 67.2% pass@128, whereas Leanabell-Prover-V2-KM reaches 68.4% pass@32 and 70.4% pass@128, yielding gains of +5.3% and +3.2%; for DeepSeek-Prover-V2-7B, the baseline is 75.6% pass@32 and 76.2% pass@128, while Leanabell-Prover-V2-DS reaches 76.6% and 78.2%, corresponding to +1.0% and +2.0%. The same study reports that vanilla RL without verifier feedback in context produces smaller or no gains, and that one round of feedback-based correction improves Kimina from 64.7% to 68.4% pass@32 and DeepSeek from 75.4% to 76.6% (Ji et al., 11 Jul 2025).

Process-Verified Reinforcement Learning for Theorem Proving via Lean evaluates whole-proof generation with STP-Lean and DeepSeek-Prover-V1.5. For STP-Lean, MiniF2F pass@32 improves from 55.9% to 57.1% and pass@64 from 56.7% to 59.2%; on ProofNet, pass@32 improves from 17.2% to 18.6%, while pass@64 changes from 19.1% to 19.0%. For DeepSeek-Prover-V1.5 + STP SFT, MiniF2F pass@32 improves from 54.9% to 56.3%, and ProofNet pass@64 improves from 17.7% to 18.5%. Ablations further report that outcome+tactic supervision outperforms outcome-only GRPO and tactic-only variants in the cited settings, and that first-token credit is consistently superior to assigning tactic-level advantage to all tokens, the last token, or entropy-based within-tactic alternatives (Kim et al., 18 Jun 2026).

In Isabelle/HOL, ProD-RL improves pass@16 from 40.8% to 45.5% on the AFP test set and from 36.5% to 39.5% on the AFP 2023 out-of-distribution set relative to ProD-SFT. During RL on AFP, newly proposed correct lemmas constitute 37.7% of the replay buffer. The paper also reports that RL without lemma proposal does not improve SFT on either test set, which isolates hierarchical decomposition and lemma proving as the operative source of gain (Dong et al., 2024).

ProofNet++ reports a broader neuro-symbolic verifier-guided pipeline rather than a single formalization of FineProofs-RL, but it is directly relevant to the same design space. Its experiments report FPSR 68.4%, PPC 81.2%, and EDPT 3.2 on miniF2F; FPSR 74.9%, PPC 88.0%, and EDPT 2.4 on mathlib-extract; and FPSR 63.5%, PPC 76.5%, and EDPT 4.0 on the HOL Light Testbed. The self-correction module yields a 36% reduction in EDPT and a 12% absolute increase in proof success rate post-correction (Ambati, 30 May 2025).

A downstream specialization of FineProofs-RL appears in ProofOptimizer, which treats proof simplification as a Lean-verified RL problem. On very long prover-generated proofs, iterative shortening reduces average proof length by 87.9% on miniF2F, 57.2% on PutnamBench, and about 49% across Seed-Prover’s IMO 2025 formal proofs P1–P5; the simplified proofs also verify faster in Lean in many cases, and supervised finetuning on simplified proofs improves miniF2F accuracy by about 2% relative to training on the original long proofs at matched loss (Gu et al., 17 Oct 2025).

5. Limitations, failure modes, and common misconceptions

A recurrent misconception is that FineProofs-RL is simply RL from a binary theorem-level success signal. That characterization is incomplete. Some systems do use sparse success/failure rewards, but others explicitly expose tactic-level local soundness, earliest failing steps, multi-turn compiler logs, or hierarchical lemma correctness. The process-verified work is explicit that outcome-only signals are sparse and provide weak guidance for long reasoning chains, while Leanabell reports that richer AST-based shaping did not outperform the simple success/failure scheme it adopted at 7B scale (Kim et al., 18 Jun 2026, Ji et al., 11 Jul 2025).

A second misconception is that more detailed reward shaping is automatically superior. The published evidence is mixed. Leanabell explored fine-grained shaping based on the verifier’s AST, including tactic efficiency, tactic power, and goal reduction, but found that the simple $$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$2 design was most effective. By contrast, process-verified RL reports gains from differentiated tactic penalties, first-error propagation, and first-token credit. The literature therefore supports no universal ranking between sparse and dense rewards; the effective choice depends on proof granularity, model scale, and the stability of token-level credit assignment (Ji et al., 11 Jul 2025, Kim et al., 18 Jun 2026).

Failure modes remain substantial. Leanabell highlights type mismatches, misuse of tactic APIs, overly long or under-specified chains of thought that never emit proper <code> blocks, sparse rewards on hard problems, and instability when verifier text is not masked out of the loss. It also reports that sample-level averaging of the GRPO-like kind underweighted long outputs and led to training collapse in that setting, whereas token-level losses avoided this. In process-verified RL, locally sound steps can still be strategically unproductive, fixed penalties show some sensitivity across models and datasets, and portability to Coq or Isabelle would require analogous tactic parsing and error semantics. ProD-RL identifies distribution shift, over-proposing irrelevant lemmas, difficulty on depth $$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$3 theorems, and verification bottlenecks from Sledgehammer/tactic timeouts; on miniF2F, its hierarchical decomposition strategy is reported as less effective than on AFP-style data (Ji et al., 11 Jul 2025, Kim et al., 18 Jun 2026, Dong et al., 2024).

Proof simplification exposes a distinct trade-off between brevity and explanatory structure. ProofOptimizer reports diversity collapse under RL, stronger red@1 than red@32, short-but-slow regressions when token reduction is optimized without heartbeat awareness, and the possibility that very short proofs hide ideas behind heavy automation such as linarith, omega, aesop, or norm_num. This indicates that FineProofs-RL objectives can optimize verifiability and compactness without necessarily optimizing human interpretability (Gu et al., 17 Oct 2025).

6. Extensions and adjacent directions

The verifier-centered logic of FineProofs-RL has already expanded beyond direct theorem proving. In autoformalization, FormaRL trains on unlabeled natural-language statements using a binary reward $$J(\theta)=\mathbb{E}_{\tau\sim\pi_\theta}\left[\sum_{t=0}^{T}\gamma^t r_t\right].$$4, where syntax correctness comes from Lean 4 compilation and semantic consistency comes from an LLM judge. Using only 859 unlabeled statements, the paper reports that Qwen2.5-Coder-7B-Instruct improves on ProofNet from 4.04% to 26.15% pass@1 and on the out-of-distribution uproof benchmark from 2.4% to 9.6% pass@1, with uproof pass@16 improving from 24.4% to 33.6% (Huang et al., 26 Aug 2025).

A second extension replaces executable formal verifiers with learned proof-checking reward models. Proof-RM constructs “question–proof–check” triplets at scale, trains a generative reward model that outputs a structured critique and verdict, and adds process reward and token weight balance to stabilize RL. Its reported in-distribution accuracies are 76.8% for 8B, 79.0% for 14B, and 82.4% for 32B, with strong generalization on OPC and ProofBench. This line of work preserves the FineProofs-RL emphasis on process-aware proof supervision, but shifts the reward oracle from a proof assistant to a scalable proof-checking model (Yang et al., 2 Feb 2026).

At the level of natural-language olympiad proofs, MaxProof combines proof generation, proof verification, and critique-conditioned proof repair in a population-level test-time scaling framework. Its merged M3 model, under MaxProof test-time scaling, reaches 35/42 on IMO 2025 and 36/42 on USAMO 2026. Although this setting is not formal theorem proving in Lean or Isabelle, it extends the same generative-verifier RL principle to long-form proof search under a defense-in-depth verifier engineered for low false-positive rate (Chen et al., 11 Jun 2026).

An important systems-level direction concerns the SFT-to-RL handoff itself. Rejuvenation studies why excessive SFT can make later RL ineffective by reducing model plasticity, and proposes base-anchored model fusion together with targeted neuron reset. The reported result is that, on math reasoning tasks, softblue+RL improves the overall average to 17.5 compared with 14.9 for softred+RL, with especially strong out-of-distribution recovery. A plausible implication is that FineProofs-RL performance depends not only on verifier design and reward shaping, but also on whether the pretrained prover remains sufficiently plastic to respond to RL updates (Liu et al., 7 Jun 2026).

Across these lines of work, the stable core of FineProofs-RL remains unchanged: proofs are optimized against feedback grounded in symbolic verification or a verifier-aligned oracle; credit assignment is pushed below the theorem level whenever possible; and the proof assistant is treated as an active component of training rather than a passive evaluation endpoint.