Abduction Prover in Isabelle/HOL

We address the use of large language models (LLMs) to help discover Isabelle proofs. An Isabelle build establishes that the submitted theory is accepted, but not that an LLM changed only what the developer authorised. We present CAPRI, a contract-aware repair workflow in which Isabelle checks the proof and an independent checker enforces a machine-readable edit contract. Prompts, proposals, candidate repositories, diagnostics, verdicts, and hashes are retained for audit. We evaluate five workflows on twelve failed proofs from four developments, with three replicates per task and condition, giving 180 runs and 138 valid repairs. Of 144 terminal candidates accepted by Isabelle, six had modified protected text; all arose in iterative workflows that could edit a complete theory. A proof-body-only interface produced 29/36 valid repairs and no contract violations, compared with 31/36 for the corresponding full-theory workflow. One-shot repair produced 22/36, while a later prospectively frozen iterative workflow produced 32/36; these figures compare complete workflows rather than individual mechanisms. A separate post hoc OpenRouter campaign found no improvement in the designated Luna comparisons. A Sol configuration with matched demonstrations produced 33/36 repairs, compared with 29/36 in the frozen OpenAI Responses condition, but the difference was not statistically significant in a one-sided exact McNemar test ($p=0.0625$).

Type annotations are essential when printing terms in a way that preserves their meaning under reparsing and type inference. We study the problem of complete and minimal type annotations for rank-one polymorphic $λ$-calculus terms, as used in Isabelle. Building on prior work by Smolka, Blanchette et al., we give a metatheoretical account of the problem, with a full formal specification and proofs, and formalize it in Isabelle/HOL. Our development is a series of experiments featuring human-driven and AI-driven formalization workflows: a human and an LLM-powered AI agent independently produce pen-and-paper proofs, and the AI agent autoformalizes both in Isabelle, with further human-hinted AI interventions refining and generalizing the development.

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.