Math Proof

Automated step-by-step proof generation and review for user-submitted theorems with iterative refinement until acceptance or retry limit reached.

🧮 math-proof — 数学定理证明与审查

Automated mathematical proof generation with built-in A/B review loop. Given a theorem statement, spawns Agent A to produce a proof and Agent B to review it. If B rejects, A revises — loop continues until B accepts or a configurable retry limit is reached.

When to Use

Activate when the user asks to prove a mathematical statement (theorem, proposition, lemma). Trigger keywords: "证明", "求证", "prove", "proof", "定理", "theorem", "推导", "demonstrate".

How It Works

text

User: "prove that the medians of a triangle are concurrent"

┌─────────────────────────────────────────────────────────┐
│  1. Ask user for configurations (feedback mode, refs)  │
│  2. Agent A produces a proof (spawned as subagent)     │
│  3. Agent B reviews the proof (spawned as subagent)    │
│  4. If B rejects → send feedback to A → revise         │
│  5. If B accepts → present final proof to user         │
│  6. Loop up to configurable max retries (default: 50)  │
└─────────────────────────────────────────────────────────┘

Pre-loop Configuration

Before spawning the first agent, ask the user for the following:

1. Feedback Mode

Mode	Description
🔍 Hint-only	Only pass direction-level hints to A (e.g., "Step 3 was rejected because it assumed what needs to be proved"). A must figure out the fix independently.
📝 With history	Direction hint + A's own previous proof text. A can base revisions on what it wrote before instead of starting from scratch.
📋 Full disclosure	Direction hint + A's previous proof + B's complete review (error locations, suggestions, fix directions). Most informative for A.

Default: Hint-only. If user doesn't specify, use hint-only.

2. Reference Materials (optional)

Ask: "Do you want A and B to reference any local files or folders? (e.g., textbook chapters, lecture notes, theorem references)"

If yes, the user specifies a path (file or folder). Its contents will be included in every spawn prompt sent to both A and B as context/reference material.

Constraints & Rules

User-Supplied Constraints

The user may supply constraints that must be passed to both A and B:

• Proof method restrictions: "纯几何" / "综合法" / "不能用向量" / "不能用塞瓦定理" / "解析几何" / "代数法" / "数学归纳法"等
• Tool restrictions: "不能用坐标" / "不能用微积分" / "初等方法"等
• Format requirements: "用 LaTeX" / "用中文" / "step by step"等

These constraints must be included verbatim in every spawn prompt sent to both A and B.

Built-In Rules (for this skill)

• A (producer): Must produce a self-contained proof following all user-supplied constraints. Proof must be complete, step-by-step, with each step justified.
• B (reviewer): Must check for logical gaps, hidden assumptions, circular reasoning, missing steps, and constraint violations. B's review must be specific: point to exact line/step numbers where issues exist.
• Loop termination: Loop ends when B marks the proof as "通过" / "passed" / "accepted", or when the retry limit is reached.

Implementation

Spawn A (first attempt)

text

{
  "mode": "run",
  "task": "请严格证明以下定理：\n\n[定理陈述]\n\n[参考材料（如有）]\n\n要求：\n[用户约束，逐条列出]\n\n用 LaTeX 格式输出完整证明。"
}

Spawn B (review)

text

{
  "mode": "run",
  "task": "请严格审查以下证明。\n\n[证明全文]\n\n[参考材料（如有）]\n\n请逐行检查：\n1. 每一步的推理是否正确？\n2. 有没有逻辑漏洞或循环论证？\n3. 有没有违反用户要求的约束（[约束列表]）？\n4. 整体逻辑链是否完整？\n5. 评价：通过 / 需要修改（说明理由）"
}

Spawn A (revision, n > 1)

Hint-only mode (默认):

text

{
  "mode": "run",
  "task": "请重新证明以下定理。之前的版本被驳回，需要修改。\n\n[定理陈述]\n\n[参考材料（如有）]\n\n你上一轮写的证明原文：\n[A上一轮的完整证明]\n\n提示：上一轮被驳回的原因是[方向性提示，不包含B的完整审查]。\n\n要求：\n[用户约束，逐条列出]\n\n用 LaTeX 格式输出完整证明。"
}

With-history mode:

text

{
  "mode": "run",
  "task": "请重新证明以下定理。之前的版本被驳回，需要修改。\n\n[定理陈述]\n\n[参考材料（如有）]\n\n你上一轮写的证明原文：\n[A上一轮的完整证明]\n\n提示：上一轮被驳回的原因是[方向性提示]。你可以参考自己之前写的证明进行修改。\n\n要求：\n[用户约束，逐条列出]\n\n用 LaTeX 格式输出完整证明。"
}

Full disclosure mode:

text

{
  "mode": "run",
  "task": "请重新证明以下定理。之前的版本被驳回，需要根据审查意见修改。\n\n[定理陈述]\n\n[参考材料（如有）]\n\n你上一轮写的证明原文：\n[A上一轮的完整证明]\n\n审查意见原文：\n[B的完整审查结果]\n\n要求：\n[用户约束，逐条列出]\n\n用 LaTeX 格式输出完整证明。"
}

Output Format

Success (B accepts)

text

🎉 证明通过！

[完整证明文本]

Failure (retry limit reached)

text

😿 已达到最大重试次数（N次），证明未能通过审查。

最后一份证明的问题：[概述]

Example Run

Theorem

三角形的三条中线交于一点（重心定理）

User Constraints

• 纯欧几里得几何方法
• 不能使用向量、坐标、解析几何
• 不能使用塞瓦定理、梅涅劳斯定理

Pre-loop Config

• Feedback mode: Hint-only
• Reference: None

Trace

Round	A Action	B Action	Result
1	向量法证明	B审查	❌ 违反约束（用了向量）
2	综合法，相似三角形	B审查	❌ 第三步逻辑漏洞
3	修正第三步	B审查	❌ 循环论证
4	平行四边形构造	B审查	❌ 未定义符号
5	新思路尝试	B审查	❌ 方向错误
6	平行四边形+比例	B审查	❌ 共线未证明
7	相似+唯一性	B审查	❌ 论证不完整
8	修正表述	B审查	❌ 根本逻辑问题
9	倍长中线法	B审查	✅ 通过

Notes

• Keep the loop summary updated between rounds so the user can see progress
• After each round, report which round (N/10) and a quick status
• For long-running proofs, consider asking the user if they want to continue after ~5 rounds
• If B's rejection is about a trivial formatting issue, fix and re-submit without counting it as a full round