ProofSketch: Automatically Generating Structured Mathematical Proof Outlines Using Reasoning Models

The ProofSketch project uses Xiaomi's MiMo reasoning model to convert informal mathematical propositions into structured proof outlines. Its core goal is to address the challenges in learning and generating mathematical proofs: traditional formalization tools (such as Lean and Coq) have a steep learning curve, while large language models produce unstable outputs with a lack of structure. By generating Abstract Syntax Trees (AST) that include strategies, steps, dependency relationships, and confidence levels, ProofSketch makes the proof process auditable and verifiable, striking a balance between full formalization and free text.

Mathematical proofs are a difficult point for learners; even simple propositions (such as the sum of two even numbers being even) require rigorous understanding of definitions, strategy selection, and step organization. Traditional proof assistants require mastery of complex formal languages, while proof texts generated by large language models have issues of unstable outputs and hidden errors. ProofSketch aims to fill the gap between full formalization and free text generation.

ProofSketch treats proof writing as a reasoning task, using Xiaomi's MiMo reasoning model to output structured ProofOutline AST instead of free text. The input is an informal proposition, and the output includes 6 proof strategies (direct proof, induction, etc.), an ordered list of steps (with explicit justifications), a step dependency graph, markers for unclosed gaps, and confidence levels. The technical architecture is: MiMo generates AST → Python deterministic renderer → Markdown/JSON output.

Taking 'the sum of two even numbers is even' as an example, the outline generated by ProofSketch uses the direct proof strategy (confidence level 0.95). The steps include: assuming a and b are even → converting to 2j and 2k based on definitions → algebraic operation to get 2(j+k) → concluding it is even. Each step is annotated with justifications (such as definitions, algebraic laws) and dependency relationships. Its features are clear strategies, ordered steps, explicit justifications, visual dependencies, and quantified confidence levels.

Comparison with existing tools:

Feature	GPT-4 Free Text	Lean/Coq	ProofSketch
Structured AST	No	Yes	Yes
Strategy Selection	Yes	N/A	Yes
Explicit Gap Marking	Partial	Yes	Yes
Step Justifications	Partial	Yes	Yes
Learning Curve	Low	High	Medium
ProofSketch is positioned as an intermediate solution; although it is not a proof checker, explicit gap marking makes errors easier to detect.

In terms of technical implementation, the project uses Pydantic to define the ProofOutline schema to ensure correct formatting. Usage methods include:

1. Command-line tool: supports Markdown/JSON output;
2. Python library: integrate to generate outlines;
3. Configure MiMo API: set the key and model via environment variables. The model explicitly marks unclosed gaps and does not hide errors.

Future plans include a Lean4 Stub renderer, step dependency graph visualization, and adversarial checking. Summary: ProofSketch helps learners understand reasoning logic by deconstructing the proof process, providing an easy-to-use structured solution for mathematics education, formal verification beginners, and researchers. Its hybrid architecture provides a reference for the design of other AI-assisted tools.