LLM-based Theorem Provers

• LLM-based theorem provers are neuro-symbolic systems that fuse transformer language models with formal proof assistants like Lean, Isabelle, and Coq for automated tactic prediction and verification.
• They employ multi-turn reinforcement learning, hierarchical planning, and retrieval-augmented techniques to surpass dataset constraints and achieve state-of-the-art performance on benchmarks such as MiniF2F and ProofNet.
• Modular pipelines and lightweight inference methods like activation steering enable dynamic subgoal refinement and efficient handling of complex proof structures.

LLM-based theorem provers constitute a class of neuro-symbolic systems that integrate autoregressive, transformer-based neural models with formal reasoning environments such as Lean, Isabelle, and Coq. These systems leverage LLMs for tactic prediction, proof planning, or synthesis, while using proof assistants’ verification machinery to ensure correctness at the level of each inference step or entire proof. The LLM’s role typically includes next-step proposal, premise selection, subgoal decomposition, conjecture generation, or even learning to invent new intermediate lemmas, bridging statistical/linguistic pattern recognition with the strict demands of proof verifiers. Recent research in this domain is marked both by algorithmic innovations in RL and search, and by integration of hybrid neural–symbolic architectures that combine data-driven reasoning with explicit formal verification.

1. Multi-Turn RL for Step Provers and Hierarchical Planning

LLM-based theorem provers originally relied on imitation learning from human proofs, but performance plateaued quickly due to dataset constraints and lack of sustained improvement under self-play. To address this, frameworks such as BFS-Prover-V2 (Xin et al., 8 Sep 2025) introduced a multi-turn off-policy RL paradigm for step-wise tactic generation in Lean4, combining AlphaZero-inspired expert iteration, adaptive tactic-level data filtering, and periodic “soft reset” retraining rounds to promote long-horizon improvement. The proof search is cast as a Markov Decision Process, where the state is a Lean proof state and the actions are discrete tactics. Each RL loop alternates between large-scale BFS tree search (generating synthetic proof trajectories) and fine-tuning (policy improvement), robustly escaping RL plateaus—performance grows monotonically for 20+ iterations only if adaptive data filtering and retraining are enforced.

At inference, scalability is achieved by planner-enhanced multi-agent search: a general-purpose LLM decomposes the theorem into intermediate subgoals, which are then solved in parallel by an array of prover agents, mediated by a shared proof-cache. Subgoal decomposition reduces search depth, while dynamic replanning allows adaptive refinement if subgoal proofs fail. This multi-agent hierarchical approach achieves state-of-the-art on MiniF2F (95.08%) and ProofNet (41.4%) with 32B-class models, exceeding strong baselines and matching dedicated end-to-end proof synthesis systems (Xin et al., 8 Sep 2025).

2. Modular Hybrid and Sketch-Based Proof Synthesis

Hybridization between whole-proof generation and step-wise tactics forms the architectural core of several recent systems. HybridProver (Hu et al., 21 May 2025) uses a dual-model pipeline for Isabelle in which a whole-proof LLM generates Isar proof candidates, and any that fail formal verification are mechanically abstracted into proof “sketches”—where all tactics are replaced by sorry. Then, a tactic-generating LLM (augmented with ATP calls like Sledgehammer) fills in each sketch’s subgoals. This decouples high-level logical planning from low-level discharge and avoids the failure modes of either class in isolation. On miniF2F, this hybrid paradigm outperforms prior SOTA by 3.3pp (59.4% vs 56.1%), with ablation studies showing that neither proof synthesis nor solo tactic refinement attains comparable reliability (Hu et al., 21 May 2025).

Relatedly, ProofAug (Liu et al., 30 Jan 2025) proposes an “automate-at-granularity” method: LLM outputs are parsed into blocks, maximal typecheckable fragments (MCSPs) are extracted, and all “sorry” gaps are independently filled using local ATPs or tactic heuristics at the minimal failing granularity, further boosted via recursive best-first search (ERP). This modularity achieves 66% cumulative pass@k with only ~2100 LLM queries per problem—lower than the 16K+ typical in previous approaches.

3. Lightweight Guidance and Activation Steering

While large LLMs possess general reasoning capacities, their behavior in tactic selection is suboptimal in resource- or training-constrained settings. Activation steering (Kirtania et al., 21 Feb 2025) proposes an “inference-only” intervention: a computed steering vector, derived from hidden state differences between successful and failed tactic predictions, is injected into an intermediate residual stream of the LLM during inference. This technique boosts MiniF2F pass rates by up to +3% over fine-tuned and retrieval-augmented models—demonstrating that lightweight, activation-space reweighting can systematically shift tactic ranking towards domain-specific reasoning paths, notably outperforming random-vector ablations (Kirtania et al., 21 Feb 2025).

This suggests steering may serve as a data-efficient substitute or complement to full model fine-tuning, especially where memory and compute are limited or for rapid adaptation.

4. Curriculum Generation, Self-Play, and Data Synthesis

Scalability of LLM-based provers fundamentally depends on access to rich, diverse proof data. To overcome the bottleneck of limited human-annotated theorems, modern systems deploy several strategies:

• Self-Play and Iterative Conjecturing (STP): STP alternates between a conjecturer and a prover, both roles instantiated by LLMs. The conjecturer proposes fresh conjectures related to recently proved theorems, which are attempted by the prover; the two are trained on barely-provable instances, ensuring the curriculum adapts to the frontier of model capability. This achieves a doubling in LeanWorkbook proof rate (13.2%→26.3%) and sets state-of-the-art on whole-proof miniF2F and ProofNet (Dong et al., 31 Jan 2025).
• Expert Iteration with Partial-Credit RL: ProD-RL (Dong et al., 2024) explicitly rewards the model for proposing and proving novel, correct lemmas during hierarchical decomposition, even if the original theorem is not yet solvable—mirroring mathematical research practice. With 37.7% of the replay buffer consisting of newly-invented lemmas and improved pass rates (45.5% on the AFP test set), this hierarchical, RL-based training regime enables robust learning in data-limited, “no-hints-provided” generalization settings.
• Synthetic Data via Proof-State Exploration: Diverse tactic transitions are generated by forced exploration of proof neighborhoods, exposing the LLM to hard and rare intermediate states, and providing high-quality, decontaminated transitions for fine-tuning policy models. In conjunction with adaptive beam search, this leads to 60.74% pass@1 on MiniF2F, topping strong baselines (Lai et al., 17 May 2025).

This line of work highlights that LLM theorem provers are bottlenecked at scale not by model size alone but equally by diversity and continual renewal of the proof search curriculum.

5. Retrieval Augmentation, Premise Selection, and Neuro-Symbolic Integration

LLM-theorem provers often struggle to identify the most relevant premises in large libraries. Retrieval-augmented frameworks such as LeanDojo (Yang et al., 2023) deploy dense retrievers trained on fine-grained premise annotations, feeding contextually filtered lemmas into the LLM for each tactic prediction. The ReProver system thus improves pass@1 over both non-retrieval LLM baselines and even GPT-4 zero-shot on mathlib benchmarks, and sets a reproducible standard for open-source, retrieval-augmented proving.

More generally, many systems hybridize neural and symbolic approaches. For example, PALM (Lu et al., 2024) implements a generate-then-repair scheme in which LLMs draft proofs but rely on symbolic repair modules (reference replacement, CoqHammer backtracking) for low-level error correction—doubling or tripling baseline proof rates. Strat2Rocq (Fang et al., 11 Oct 2025) takes offline LLM-generated proof trajectories, distills common strategies as new verified lemmas, and integrates these into symbolic ATP-based approaches (CoqHammer), yielding a 13.41% increase in automatic proof rates on large verification projects.

Neuro-symbolic fusion thus forms both a performance driver and a path towards system explainability and formal trustworthiness.

6. Specialization for Hierarchical and Domain-Specific Provers

LLM-based theorem proving techniques extend to settings with specialized proof structures and logics:

• Hierarchical Decomposition in Non-Tactic Provers: In TLA+, the lack of tactic corpora and hierarchical proof format motivates claim-decomposition methods where LLMs generate only normalized sub-claims subject to rigid grammar constraints, while symbolic provers discharge each as independent proof obligations. This approach (LMGPA) achieves 59.1% on translated miniF2F/ProofNet while keeping syntactic validity high (Zhou et al., 10 Dec 2025).
• First-Order Logic (FOL) Deduction: DREAM (Cao et al., 20 Jun 2025) addresses combinatorial failures in FOL multi-step theorem provers by enforcing axiom-driven strategy diversification and sub-proposition error feedback, systematically exploring alternative logical branches and attributing errors to specific sub-proofs. DREAM yields 6–8pp improvements on strict FOL benchmarks, where state-of-the-art LLMs otherwise falter below 5% (Cao et al., 20 Jun 2025).
• Automated Reasoning for Natural-Language Proofs: DeepTheorem (Zhang et al., 29 May 2025) augments LLM informal theorem proving with a 121K-sample IMO-level corpus and RL-Zero variant rewards, achieving competitive or superior accuracy to much larger models on mathematical reasoning benchmarks, demonstrating that natural-language alignment remains a strong path for advancing LLM mathematical fluency.

7. Challenges, Limitations, and Outlook

Notwithstanding rapid advances, the field confronts multiple challenges:

• Search and Computation: Hierarchical and multi-agent search architectures, while powerful, are bottlenecked by planner latency at scale (beyond 16 parallel agents in BFS-Prover-V2) (Xin et al., 8 Sep 2025). Proof search often remains combinatorially hard, particularly in settings with deep or highly branching proof trees.
• Data Scarcity and Generalization: Systems reliant on human-annotated proofs face coverage gaps and rapidly saturate. Where synthetic or self-play data is used, domain transfer remains limited—hierarchical RL gains vanish on OOD benchmarks with different proof logic (Dong et al., 2024).
• Context Handling and Retrieval: Fixed-length context windows limit the number of premises deliverable to the LLM; scalable selection and flexible context windows are active areas of investigation (Yang et al., 2023).
• Inference-Phase Engineering vs. Training: While lightweight approaches such as activation steering are effective in low-resource regimes, they may be insufficient for very heterogeneous domains or deep proofs (Kirtania et al., 21 Feb 2025).

A plausible implication is that the most robust, scalable LLM-based theorem provers will integrate sustained RL loops, modular proof decomposition, retrieval-augmented premise selection, synthetic curriculum generation, and lightweight inference-time guidance within a unified neuro-symbolic architecture—and that generalization across formal systems and mathematical domains will demand richer, cross-disciplinary benchmarks alongside model advances.