AoA: Theorem Proving Agent over Abstract Syntax Tree of Redesigned Language

2026-07-17

AI Summary

This work addresses the limitations of existing large language model–driven theorem-proving agents, which rely on concrete syntax and consequently incur high token consumption, elevated API costs, and poor adaptability to new proof languages such as Minilang. To overcome these issues, the authors propose the first proof agent built upon abstract syntax trees (ASTs), generating Minilang ASTs in JSON format and introducing a tree-editing model that unifies proof actions and subgoal states within a single proof tree, thereby eliminating dependence on source code text and line numbers. The approach achieves substantial efficiency gains: on the miniF2F and NTP4VC-Pearl benchmarks, it reduces token usage by 2.9–6.9×, lowers API costs by 2.3–4.7×, decreases tool calls by 3.9–8.9×, accelerates reasoning by 1.4–2.0×, and solves more problems on challenging verification tasks.

Abstract

Interactive theorem proving (ITP) underpins program verification and formalized mathematics, but its manual effort limits scalability. LLM-based proof agents promise to ease this effort, but their heavy token consumption and API cost remain a major obstacle. We trace this cost to a shared root: current agents operate on serialized concrete syntax, emitting proofs as source text and recovering proof states through separate, line-number-based queries, so every edit shifts later lines and forces repeated relocation of errors and states. This same dependence on concrete syntax also blocks adoption of Minilang, a recent proof language that reaches SOTA on LLM-based proving but is too new for LLMs' training corpora. We address both problems by lifting the agent off source text and onto the abstract syntax tree (AST): the model supplies proofs as JSON representations of Minilang's AST -- native to tool-calling LLMs -- and drives the prover through a tree-edit model that fuses proof operations and states into one proof tree, so each operation carries its own subgoal's state, readable directly off the tree. We realize this design in \emph{Agent over AST} (AoA). Against Amazon's Isabelle Agent on miniF2F and NTP4VC-Pearl common success sets, AoA cuts API cost by 2.3--4.7x (normalized input-cache accounting), uses 2.9--6.9x fewer tokens and 3.9--8.9x fewer tool calls, and finishes 1.4--2.0x faster -- while also solving far more problems on the harder verification benchmark.

Problem

• interactive theorem proving
• concrete syntax
• API cost
• token consumption
• proof language

Innovation

• Abstract Syntax Tree
• Interactive Theorem Proving
• Minilang
• Tree-edit Model
• LLM-based Proof Agent