Characterizing initial human-AI proof formalization workflows

Abstract: For centuries, human mathematicians have written proofs to substantiate their mathematical arguments; yet, the ability to automatically verify the validity of proofs has long been a challenge. Advances in AI systems' ability to generate code and engage in increasingly high-level mathematical reasoning promise to transform people's ability to formalize and thereby verify proofs. While many works focus on benchmarking the current frontier, we instead study how people use these tools. We conduct a mixed-methods analysis into the initial impact of AI on people's formalization workflows: what people claim they want, what they see as the barriers to those visions, and how they actually use and adapt AI in practice. A qualitative survey shows that people's preferences are diverse, but with a general desire for AI assistance in formalization that preserves high-level human control over the proof discovery process. To assess how people actually engage with AI for formalization under such limitations, we conduct a controlled user study in which participants formalize informal math problems and their proofs, with and without AI, across a range of mathematical problems at varying levels of difficulty and domains. Despite limitations of the tools at the time for autoformalization, participants tend to attain higher formalization accuracy when allowed access to AI tools than when formalizing on their own, with most participants flexibly choosing to use multiple different AI tools. Taken together, our work sheds light on the early stages of AI integration into formalization workflows, involving an intimate interplay of human and AI engagement.

• The paper combines a survey of 31 respondents with a controlled study of 7 Lean users to examine preferences, barriers, and real-world human–AI formalization workflows.
• AI tool access increased correct formalizations by 0.30 points (95% CI: 0.10–0.50; p = 0.003) but did not significantly reduce completion time, suggesting tools improve solvability more than speed.
• Most participants combined multiple tools for lemma search, planning, autocomplete, and automated proving, while 81.7% of survey respondents preferred full or partial human control and reported reliability and integration problems.

This paper presents a mixed-methods investigation into how mathematicians and mathematically trained users integrate AI tools into proof formalization workflows in Lean (2606.04273). Rather than benchmarking model capabilities, the authors study preferences, perceived barriers, and observed practice through a qualitative survey (31 respondents after quality filtering) and a controlled user study (7 participants, over 80 hours of screen-recorded formalization). The work provides an early snapshot of human-AI interaction in formal mathematics, conducted before the widespread adoption of agentic coding systems.

Survey: heterogeneous preferences with a shared desire for control

The survey, run between October 27 and November 25, 2025, recruited respondents via mailing lists and social media; nine of forty responses were excluded for lacking formalization experience or substance. Self-reported AI use varied widely and was largely independent of Lean community engagement (mathlib contribution status) or prior experience.

Three quantitative patterns stand out:

• Automation appetite is selective: 70.3% of respondents wanted AI to automate core aspects of formalization, but only 40.7% mentioned AI use in proof discovery, and only 33.3% of those discussing formalization wanted the entire process automated.
• Control dominates: 81.7% preferred full or partial human control; 66.7% wanted to retain high-level creative or strategic control, while 14.8% said they would never want to use AI at all.
• Reliability frustration coexists with continued use: 73.3% of respondents who sometimes or always use AI cited lack of reliability as their dominant frustration, and 18.5% raised formalization-specific complaints (style mismatch, poor integration into existing workflows).

The implication is that demand for AI in formalization centers on mechanical assistance rather than autonomous proof generation, and that evaluation efforts targeting only formal correctness miss concerns about style and workflow fit that matter to practitioners.

User study design

Seven participants (from eight; one excluded for missing video recordings), all with baseline Lean familiarity and at least undergraduate mathematics training—roughly split between "Beginner" ($N=3$) and "Advanced/Expert" ($N=4$)—formalized six informal problems spanning number theory to topology, each including both statement and full proof. Problems were split into two difficulty-matched groups of three across two weeks, with tool access varying by week (randomized order): one "human alone" week (editor and compiler only) and one week with unrestricted access to AI tools such as ChatGPT, Claude, GitHub Copilot, LeanSearch, and Lean Copilot. Accuracy was graded manually by experienced Lean users as binary correctness of statement plus proof; time was measured from screen recordings.

Tool access improves accuracy but not speed

In a linear mixed-effects model with crossed random intercepts for participant and problem, tool access increased the probability of a correct formalization by $0.30$ points (95% CI $[0.10, 0.50]$; $z = 2.94$, $p = 0.003$; $n = 42$ trials from 7 participants). Notably, the random-intercept standard deviations were $0.28$ for problem versus $0.20$ for participant, indicating problem difficulty contributed more variance than individual skill. One problem ("Problem G") was solved by no participant even with AI assistance.

By contrast, tool access had no statistically significant effect on time-to-formalize (estimated effect $6.61$, 95% CI $N=4$0; $N=4$1; $N=4$2 trials). This dissociation—accuracy gains without time savings—suggests participants used AI to reach solutions they otherwise could not, rather than to work faster on problems they could already solve. The authors also concede that instructing participants to use tools in one week may have biased them toward AI use they would not otherwise have chosen.

Multi-tool workflows and active human oversight

Six of seven participants used more than one AI-based tool, deliberately matching tools to subtasks: semantic search engines (LeanSearch, Moogle) for lemma retrieval, inline LLMs (GitHub Copilot) for boilerplate and autocomplete, chatbots (ChatGPT, Claude, Gemini) for planning and whole-proof attempts, and specialized provers (Kimina Prover) for offloading easy lemmas. Video analysis revealed consistent deliberate judgment about accepting or rejecting suggestions, aligning with survey-expressed preferences for strategic control.

The authors identify three workflow archetypes:

Archetype	Participants	Character
Human formalizers with AI assistance	5	Light autocomplete and lemma retrieval; humans do most formalization
AI formalization with human help	1	Heavy AI delegation, occasional manual edits
Hybrid human-AI formalization	1	Varied multi-tool use with extensive human engagement

Two observations qualify these patterns. First, the single heavy-AI-delegation participant had limited Lean and mathematics experience and solved only one of six problems, hinting that full delegation may correlate with weaker outcomes, though the sample cannot support such a claim. Second, the hybrid participant's workflow required substantial switching among tools, which conflicts with the ease-of-use desires several survey respondents expressed—an integration gap that current tooling does not close.

Limitations and open questions

The paper is explicit about its constraints. Both studies involve few participants drawn non-systematically (survey via X and mailing lists; study participants mostly post-graduate level rather than expert mathematicians), so generalization across the mathematics community is unestablished. Accuracy grading is binary and inherently subjective under the current definition. The task isolates formalization of given informal proofs, not proof discovery itself. Coding of process-level video data carries substantial analytical overhead, and the exploratory video analyses cover only a subset of recordings. Finally, because both studies predate agentic coding systems, the workflows observed are manually assembled; whether agentic orchestration changes the control-versus-delegation trade-off people prefer remains an open question the paper poses but does not answer.

Conclusion

This work shifts attention from what AI formalization tools can do in isolation to how people actually deploy them. Its central empirical contributions are a quantified preference structure favoring human strategic control (81.7%), evidence that tool access raises formalization accuracy by $N=4$3 points despite contemporaneous tool limitations, and documentation of flexible multi-tool workflow construction as the dominant adaptation strategy. The reliability frustrations voiced by active AI users, together with style- and integration-specific complaints, indicate that practitioner-centered evaluation beyond formal correctness is needed for AI tools to fit mathematical practice as mathematicians wish to conduct it.