Last Updated: January 16, 2026 Scope: Autoformalization and formal theorem proving systems utilizing Lean 4
Achievement: Silver Medal at IMO 2024 | Nature Publication November 2025
AlphaProof is an AlphaZero-inspired agent that learns to find formal proofs through reinforcement learning by training on millions of auto-formalized problems. For the most difficult problems, it uses Test-Time RL, a method of generating and learning from millions of related problem variants at inference time to enable deep, problem-specific adaptation.
Solved 3 of 5 non-geometry problems at IMO 2024
Successfully solved Problem 6 (the hardest problem, solved by only 5 of 609 human participants)
Combined with AlphaGeometry 2, achieved silver medal equivalent (28/42 points)
Trained on 80 million formal math problems using reinforcement learning
Produces outputs in Lean formal language with automatic verification
Couples a pre-trained language model with the AlphaZero reinforcement learning algorithm
Fine-tuned Gemini model to automatically translate natural language problems into formal statements
Uses "product node" concept in search algorithm for handling proof sub-goals
Employs Test-Time RL for generating problem variants during inference
Nature Paper: https://www.nature.com/articles/s41586-025-09833-y
DeepMind Blog: https://deepmind.google/blog/ai-solves-imo-problems-at-silver-medal-level/
Achievement: Strong Prime Number Theorem Formalization | September 2025
Using Gauss, Math Inc. completed a challenge set by Fields Medallist Terence Tao and Alex Kontorovich in January 2024 to formalize the strong Prime Number Theorem (PNT) in Lean. This formalization, which had stalled after 18+ months of human expert effort, was completed in just three weeks.
~25,000 lines of Lean code produced
1,100+ theorems and definitions formalized
Completed in 3 weeks vs. 18+ months of human effort
Built substantial complex analysis infrastructure (Borel-Carathéodory theorem, etc.)
Autoformalization agent designed for assisting human expert mathematicians
Works autonomously for hours at a time
Uses thousands of concurrent agents (up to 12 hours each)
Powered by Morph Labs' Trinity/Infinibranch infrastructure
Consumes multiple terabytes of cluster RAM for Lean verification environments
"The translation of human mathematics into verifiable machine code has long been a grand challenge. However, the cost of doing so is prohibitive, requiring scarce human expertise."
This represents the first major demonstration of AI dramatically compressing formalization timelines for research-level mathematics.
Math Inc. Announcement: https://www.math.inc/gauss
GitHub Repository: https://github.com/math-inc/strongpnt
Blueprint Documentation: https://math-inc.github.io/strongpnt/
Achievement: Gold Medal at IMO 2025 | State-of-the-Art Benchmarks
Seed-Prover is a lemma-style whole-proof reasoning model that can iteratively refine proofs based on Lean feedback, proved lemmas, and self-summarization. It achieved gold-medal performance at IMO 2025 with formally verified solutions.
5/6 IMO 2025 problems fully solved with formal verification
78.1% of formalized past IMO problems solved
99.6% on MiniF2F benchmark (saturated)
>50% on PutnamBench (state-of-the-art)
11/12 Putnam 2025 problems solved within 9 hours (Seed-Prover 1.5)
Generated Lean proof code for first 5 IMO 2025 problems in 16.5 hours
Introduced Agentic Prover architecture with:
Mathlib Search Tool
Python Code Execution
Incremental Lemma Verification
Uses Sketch Model for proof decomposition via Rubric RL
Lemma-style proving as core methodology
Three specialized test-time inference strategies for deep and broad reasoning
Seed-Geometry engine for geometry problems (solved IMO P2 in 2 seconds)
Multi-stage, multi-task RL based on VAPO
Iterative refinement using Lean compiler feedback
arXiv Paper (Seed-Prover): https://arxiv.org/abs/2507.23726
arXiv Paper (Seed-Prover 1.5): https://arxiv.org/pdf/2512.17260
GitHub Repository: https://github.com/ByteDance-Seed/Seed-Prover
Achievement: Gold Medal at IMO 2025 with Formal Verification | October 2025
Aristotle is an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 IMO problems. Uniquely, all solutions were automatically verified in Lean 4—no human checking required.
5/6 IMO 2025 problems solved with formally verified proofs
Solutions verified down to foundational axioms
Demonstrated contributions to Mathlib and proficiency in category theory/homological algebra
200B+ parameter transformer backbone
Lean Proof Search System — Monte Carlo Graph Search (MCGS) over Lean 4 proof states
Informal Reasoning System — Generates and formalizes lemmas (autoformalization)
Dedicated Geometry Solver (Yuclid) — Specialized for IMO geometry problems
Hypergraph construction to identify equivalent states and actions
Test-time training from inference trace feedback
Parallelized lemma generation and tactic search
Favorable scaling properties for automated theorem proving
"When our system outputs the proof, nobody has to look at it. You know it's correct." — Tudor Achim, CEO
This addresses the verification problem: as AI generates more mathematical content, human verification becomes a bottleneck.
arXiv Paper: https://arxiv.org/abs/2510.01346
Full Paper PDF: https://harmonic.fun/pdf/Aristotle_IMO_Level_Automated_Theorem_Proving.pdf
GitHub (IMO Proofs): Referenced at https://harmonic.fun/news
Achievement: State-of-the-Art Neural Theorem Proving | April 2025
DeepSeek-Prover-V2 is an open-source large language model designed for formal theorem proving in Lean 4, using a recursive theorem proving pipeline powered by DeepSeek-V3.
88.9% pass ratio on MiniF2F-test
49/658 problems solved on PutnamBench
6/15 AIME 2024-25 problems solved
Available in 7B and 671B parameter versions
Recursive proof search with DeepSeek-V3 for subgoal decomposition
Cold-start training from synthetic data (no human-annotated proofs)
Combines informal chain-of-thought with formal Lean proofs
Uses GRPO (Group Relative Policy Optimization) for reinforcement learning
Consistency reward for structural alignment between sketch and solution
325 formalized problems for evaluation
Includes 15 problems from AIME 2024-25
Covers textbook examples and educational tutorials
arXiv Paper: https://arxiv.org/abs/2504.21801
GitHub Repository: https://github.com/deepseek-ai/DeepSeek-Prover-V2
HuggingFace Models: Available for both 7B and 671B versions
Achievement: State-of-the-Art Open-Source Theorem Prover | August 2025
Goedel-Prover-V2 is a series of open-source language models from Princeton University that set a new state-of-the-art in automated theorem proving for Lean 4. The system achieves remarkable results with models 80× smaller than competing systems.
88.1% on MiniF2F at pass@32 (standard mode)
90.4% on MiniF2F with self-correction mode
86 problems solved on PutnamBench at pass@184 (first place among open-source models)
Outperforms DeepSeek-Prover-V2-671B despite being 80× smaller (8B model)
29.7K problems solved in Lean Workbook (nearly doubling prior 15.7K)
Three key innovations:
Scaffolded Data Synthesis — Generates synthetic tasks of increasing difficulty using Lean's extract_goal to mine failing subgoals from incorrect proofs
Verifier-Guided Self-Correction — Iteratively revises proofs using Lean compiler feedback through long chain-of-thought prompts
Model Averaging — Merges model checkpoints to maintain output diversity in later training stages
| Model | MiniF2F pass@32 | PutnamBench |
|---|---|---|
| Goedel-Prover-V2-8B | 84.6% | — |
| Goedel-Prover-V2-32B | 88.1% (90.4% w/ self-correction) | 86 problems |
arXiv Paper: https://arxiv.org/abs/2508.03613
GitHub Repository: https://github.com/Goedel-LM/Goedel-Prover-V2
Earlier Version (V1): https://arxiv.org/abs/2502.07640
Achievement: First Large Formal Reasoning Model with Scaling Laws | April 2025 (72B: July 2025)
Kimina-Prover pioneers a novel reasoning-driven exploration paradigm for formal theorem proving in Lean 4. It was the first system to demonstrate clear performance scaling with model size in neural theorem provers—a property previously unobserved in formal mathematics.
80.7% on MiniF2F with pass@8192 (April 2025 release)
92.2% on MiniF2F with full Test-Time RL (TTRL) search framework
84.0% at pass@32, 86.4% with error correction, 87.7% at pass@1024
52.94% at pass@1 (one sample per problem)—exceptional sample efficiency
First to achieve 80%+ pass rate on miniF2F among published results
Formal Reasoning Pattern — Structured reasoning style that bridges informal intuition and formal verification
Test-Time Reinforcement Learning (TTRL) — Enables discovery, combination, and reuse of intermediate lemmas autonomously
Negation Proving Process — Discards logically inconsistent lemmas by attempting to prove their negation
32K token context window — Longest context used in neural theorem proving
First demonstration that larger LLMs yield stronger formal reasoning when equipped with appropriate RL pipelines:
| Model Size | MiniF2F pass@32 |
|---|---|
| 1.5B (distilled) | ~60% |
| 7B (distilled) | ~68% |
| 72B | 84.0% |
arXiv Paper: https://arxiv.org/abs/2504.11354
GitHub Repository: https://github.com/MoonshotAI/Kimina-Prover-Preview
HuggingFace Blog: https://huggingface.co/blog/AI-MO/kimina-prover
Achievement: Simple Best-First Search Achieves SOTA | February 2025
BFS-Prover challenges the prevailing perception that complex Monte Carlo Tree Search (MCTS) and value functions are necessary for large-scale automated theorem proving. It demonstrates that simple Best-First Search, with targeted optimizations, can outperform sophisticated approaches.
72.95% on MiniF2F-test (cumulative evaluation)—new SOTA at release
70.83% at fixed tactic generation budget (2048 × 2 × 600 inference calls)
Outperformed InternLM2.5-StepProver (65.9%), HunyuanProver (68.4%), and DeepSeek-Prover-V1.5 (63.5%)
Solved multiple IMO problems: imo_1959_p1, imo_1960_p2, imo_1962_p2, imo_1964_p2, imo_1983_p6
No critic model or MCTS required
Three key innovations in the expert iteration framework:
Strategic Data Filtering — Excludes problems solvable via beam search node expansion at each iteration round, focusing training on harder cases
DPO from Compiler Feedback — Applies Direct Preference Optimization using state-tactic pairs automatically annotated with Lean compiler error feedback
Length Normalization — Improves BFS search efficiency by normalizing proof length scores
"This result proves for the first time that, with a sound optimization strategy, a simple BFS method can outperform mainstream sophisticated search algorithms such as MCTS and value functions."
arXiv Paper: https://arxiv.org/abs/2502.03438
ByteDance Seed Blog: https://seed.bytedance.com/en/blog/seed-research-new-sota-in-formal-mathematical-reasoning-bfs-prover-model-now-open-sourced
Achievement: Guided Tree Search with Critic Models | December 2024 / March 2025
HunyuanProver is a language model fine-tuned from Hunyuan 7B for interactive automatic theorem proving with Lean 4. It introduced scalable data synthesis and guided tree search algorithms for effective "System 2 thinking."
68.4% on MiniF2F-test (SOTA at release, surpassing prior 65.9%)
Proved 4 IMO statements in miniF2F-test: imo_1960_p2, imo_1962_p2, imo_1964_p2, imo_1983_p6
Open-sourced 30K synthesized instances with natural language questions, formalized statements, and proofs
Two core modules:
Scalable Prover-Data Generator
Uses only open-source data for initial autoformalizer and prover training
Autoformalizer converts natural language math questions to Lean 4 format
Prover iteratively improved on generated data
Guided Tree-Search Algorithms
Distance Critic Guidance (DCG) for state selection
Process Reward Model (PRM) for proof quality estimation
Multiple critic models enable "slow thinking" essential for complex theorem proving
Explicitly trained critics for tree-search guidance are helpful
Scale of fine-tuning data for theorem proving is critical
Data curation and selection matter when sufficient training data exists
arXiv Paper: https://arxiv.org/abs/2412.20735
Achievement: Critic-Guided Search for Deeper Proofs | October 2024 (ICML AI4Math July 2025)
InternLM2.5-StepProver advances automated theorem proving by introducing a prover-critic framework that captures preference information from search trajectories to guide proof search at runtime, enabling discovery of deeper proofs.
65.9% on MiniF2F-test (SOTA at release)
27.0% pass@256 on ProofNet (surpassing DeepSeek-Prover-V1.5-RL's 25.3%)
6 problems solved on Putnam benchmark
Significant improvement from critic guidance: 59.4% → 65.9%
Trained with >20,000 CPU days of expert iteration
Critic Model — Estimates "value" of each proof state and decides which states to explore
Best-First + Critic-Guided (BF+CG) — Combines traditional BFS with critic guidance
Expert Iteration on Lean-Workbook — Large-scale training on 140K+ formalized problems
Log-linear scaling trends — Discovered relationship between solved problems, proof length, and CPU usage
"The tree structure constructed during the search process is naturally suited for process supervision. The search trajectories inherently provide the data needed to train the supervision model."
arXiv Paper: https://arxiv.org/abs/2410.15700
OpenReview (AI4Math@ICML): https://openreview.net/forum?id=qwCqeIg5iI
HuggingFace Model: https://huggingface.co/internlm/internlm2_5-step-prover
Achievement: Data-Efficient Autoformalization via RL | COLM 2025
FormaRL is a reinforcement learning framework for autoformalization that requires only a small amount of unlabeled data—addressing the critical data scarcity problem in formal verification.
4-6× improvement in pass@1 autoformalization accuracy
ProofNet: 4.04% → 26.15% with only 859 unlabeled samples
uproof: 2.4% → 9.6% (pass@1), 24.4% → 33.6% (pass@16)
Outperforms supervised fine-tuning (SFT) approaches
Integrates syntax check from Lean compiler
Uses consistency check from LLMs for semantic verification
Employs GRPO algorithm for model updates
Combines both signals as reward for RL training
5,273 proof problems from 14 classical textbooks
Covers undergraduate-level advanced mathematics
Designed to evaluate out-of-distribution autoformalization
OpenReview (COLM 2025): https://openreview.net/forum?id=Z2El1U94bq
arXiv Paper: https://arxiv.org/abs/2508.18914
GitHub Repository: https://github.com/THUNLP-MT/FormaRL
| System | Organization | Key Achievement | Date | Primary Link |
|---|---|---|---|---|
| AlphaProof | DeepMind | Silver IMO 2024, Nature publication | Nov 2025 | Nature |
| Gauss | Math Inc. | Strong PNT formalization in 3 weeks | Sept 2025 | Math Inc. |
| Seed-Prover | ByteDance | Gold IMO 2025 (formal), 99.6% MiniF2F | Aug 2025 | arXiv |
| Aristotle | Harmonic | Gold IMO 2025 (formal, auto-verified) | Oct 2025 | arXiv |
| DeepSeek-Prover-V2 | DeepSeek | 88.9% MiniF2F, 49/658 Putnam | April 2025 | arXiv |
| Goedel-Prover-V2 | Princeton | 90.4% MiniF2F, 86 PutnamBench (open-source SOTA) | Aug 2025 | arXiv |
| Kimina-Prover | Moonshot AI | 92.2% MiniF2F, first scaling laws | Apr/Jul 2025 | arXiv |
| BFS-Prover | ByteDance | 72.95% MiniF2F via simple BFS | Feb 2025 | arXiv |
| HunyuanProver | Tencent | 68.4% MiniF2F, guided tree search | Mar 2025 | arXiv |
| InternLM2.5-StepProver | Shanghai AI Lab | 65.9% MiniF2F, critic-guided search | Oct 2024 | arXiv |
| FormaRL | Tsinghua | 4-6× accuracy with 859 unlabeled samples | Aug 2025 | OpenReview |
The International Mathematical Olympiad has emerged as the definitive benchmark for measuring AI mathematical reasoning capabilities. In 2025, multiple independent systems achieved gold-medal equivalent performance with formally verified Lean 4 proofs.
Systems producing Lean 4 proofs offer a crucial advantage: no human verification required. As Tudor Achim (Harmonic) noted: "When our system outputs the proof, nobody has to look at it. You know it's correct."
Nearly all major advances use RL with formal verification as the reward signal:
AlphaProof: AlphaZero-style RL
Seed-Prover: VAPO-based multi-task RL
DeepSeek-Prover-V2: GRPO reinforcement learning
FormaRL: GRPO with Lean compiler feedback
Kimina-Prover: Large-scale RL pipeline
Goedel-Prover-V2: Expert iteration with RL
BFS-Prover demonstrated that simple Best-First Search, with careful optimization, can outperform complex MCTS and value function approaches—challenging assumptions about required algorithmic complexity.
Kimina-Prover provided the first evidence that larger models yield stronger formal reasoning capabilities when equipped with appropriate RL pipelines—a scaling property previously unobserved in neural theorem provers.
FormaRL shows 4-6× improvements with <1000 unlabeled samples
Synthetic data generation eliminates need for human annotation
Goedel-Prover's scaffolded data synthesis enables self-improving training
Gauss (Math Inc.) demonstrates AI can produce 25,000+ lines of formal Lean code
Goedel-Prover solved 29.7K Lean Workbook problems
This approaches the scale of major human formalization projects
| Benchmark | Description | Problems |
|---|---|---|
| MiniF2F | Cross-system Olympiad-level mathematics | 488 |
| PutnamBench | Undergraduate competition (Putnam 1962-2023) | 644 |
| ProverBench | DeepSeek's benchmark including AIME | 325 |
| Lean Workbook | Large-scale auto-formalized problems | 140K+ |
| uproof | Undergraduate-level textbook problems | 5,273 |
Lean 4: Primary formal verification language for all systems
Mathlib: Lean's mathematical library (~2M lines, 350K+ theorems)
LeanDojo / LeanDojo-v2: Data extraction and AI-assisted proving framework
Morph Cloud / Trinity: Infrastructure for scaling formal verification agents