SITA: A Framework for Structure-to-Instance Theorem Autoformalization

2025-11-13

AI Summary

Automating the formalization of research-level mathematical theorems in the Lean proof assistant remains challenging due to the gap between abstract mathematical structures and their concrete instantiations. Method: We propose a structured, template-driven approach that bridges this gap systematically. It employs reusable, modular templates to explicitly encode mappings from abstract structures to concrete instances; leverages large language models to generate candidate definitions and theorems; utilizes Lean’s type-class mechanism for automatic instance resolution; and incorporates structural hypothesis verification and feedback-guided iterative refinement to ensure formal correctness. Contribution/Results: This work achieves the first end-to-end automated formalization of theorems across multiple concrete instances derived from a single abstract structure. Evaluated on an optimization-theory dataset, our method successfully generated multiple correct, machine-verifiable Lean proofs. It significantly improves both the efficiency and breadth of mathematical formalization, advancing scalable, reliable automation in interactive theorem proving.

Technology Category

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

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

While large language models (LLMs) have shown progress in mathematical reasoning, they still face challenges in formalizing theorems that arise from instantiating abstract structures in concrete settings. With the goal of auto-formalizing mathematical results at the research level, we develop a framework for structure-to-instance theorem autoformalization (SITA), which systematically bridges the gap between abstract mathematical theories and their concrete applications in Lean proof assistant. Formalized abstract structures are treated as modular templates that contain definitions, assumptions, operations, and theorems. These templates serve as reusable guides for the formalization of concrete instances. Given a specific instantiation, we generate corresponding Lean definitions and instance declarations, integrate them using Lean's typeclass mechanism, and construct verified theorems by checking structural assumptions. We incorporate LLM-based generation with feedback-guided refinement to ensure both automation and formal correctness. Experiments on a dataset of optimization problems demonstrate that SITA effectively formalizes diverse instances grounded in abstract structures.

Problem

• Autoformalizing theorems from abstract structures to concrete instances
• Bridging abstract mathematical theories with Lean proof assistant applications
• Ensuring formal correctness through feedback-guided refinement with LLMs

Innovation

• Modular templates bridge abstract structures and instances
• Typeclass mechanism integrates definitions and instance declarations
• LLM generation with feedback ensures automation and correctness