Nazrin: Atomic Tactics for Graph Neural Networks for Theorem Proving in Lean 4

2026-02-21

AI Summary

This work addresses the inefficiency of existing machine-assisted theorem proving methods in searching for effective sequences of proof strategies. To overcome this limitation, the authors propose Nazrin, a novel theorem prover that unifies the proof process as a learnable task on graph structures by introducing a finite set of atomic tactics, an expression graph (ExprGraph) representation, and a transposition-atomization algorithm. The approach leverages graph neural networks to predict promising proof strategies and is implemented within Lean 4. It enables efficient training and inference on consumer-grade hardware and demonstrates strong empirical performance on benchmarks from the Lean standard library and Mathlib, significantly lowering the barrier to automated formal theorem proving.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In Machine-Assisted Theorem Proving, a theorem proving agent searches for a sequence of expressions and tactics that can prove a conjecture in a proof assistant. In this work, we introduce several novel concepts and capabilities to address obstacles faced by machine-assisted theorem proving. We first present a set of \textbf{atomic tactics}, a small finite set of tactics capable of proving any provable statement in Lean. We then introduce a \textbf{transposing atomization} algorithm which turns arbitrary proof expressions into a series of atomic tactics. We next introduce the \textbf{ExprGraph} data structure, which provides a succinct representation for Lean expressions. Finally, we present the \textbf{Nazrin Prover}, a graph neural network-based theorem proving agent using atomic tactics and ExprGraph. Nazrin circumvents many challenges faced by existing proving agents by exclusively dispatching atomic tactics, and it is robust enough to both train and evaluate on consumer-grade hardware. We demonstrate the potential of tools like Nazrin using theorems from Lean's standard library and from Mathlib.

Problem

• Machine-Assisted Theorem Proving
• Graph Neural Networks
• Lean 4
• Tactic Synthesis
• Formal Verification

Innovation

• atomic tactics
• transposing atomization
• ExprGraph
• graph neural network
• theorem proving