# Prove Math Proposition

> Prove Research-Level Mathematical Propositions

- Skill: `jiajunma/prove-math-proposition` (Agent Skill)
- Install (CLI): `npx skillmds@latest add jiajunma/prove-math-proposition`
- Raw SKILL.md: https://api.skillmd.com/api/skills/jiajunma/prove-math-proposition/raw
- Safety review: pending
- Works with: Claude Code, Claude.ai, OpenAI Codex
- Category: Research & Search
- Author: jiajunma (https://skillmd.com/u/jiajunma)
- Updated: 2026-09-22
- Page: https://skillmd.com/skills/jiajunma/prove-math-proposition

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# Prove Research-Level Mathematical Propositions

Three-step workflow: generate proof → verify (report) → review report → accept, correct, request hints, modify proposition, or identify falsehood.

## Core Principles

- **Rigor is paramount.** Every step must be logically sound and clearly explained. A correct conclusion from flawed reasoning is a failure.
- **Honesty about completeness.** If a complete proof is not found, present only significant partial results that you can rigorously prove (e.g., key lemmas, resolved cases, critical properties).
- **Consider falsity when proof is difficult.** When a proof is hard to find or repeatedly fails verification, **simultaneously** consider the possibility of counterexamples or that the proposition itself may be false. Actively search for counterexamples or reasons the statement could fail.
- **Use TeX** for all mathematics: `$...$` for inline, `$$...$$` for display.

## Output Format

Structure the response into these sections, in order:

### 1. Summary

**a. Verdict**
- Complete: "I have successfully proved the proposition. The conclusion is..."
- Partial: "I have not found a complete proof, but I have rigorously proved that..."

**b. Method Sketch**
- High-level strategy and logical flow
- Full, precise statements of key lemmas or major intermediate results
- Key constructions or case splits, if applicable

### 2. Detailed Solution

Full step-by-step proof. Each step must be logically justified. No internal commentary, alternative approaches, or failed attempts. Detail sufficient for an expert to verify without filling gaps.

## Workflow

### Step 1: Generate Proof

Produce a proof following the Output Format above (Summary + Detailed Solution).

### Step 2: Verify Proof

Act as an IMO grader. Produce a **Verification Report**:

- **Final Verdict:** "The solution is correct" / "The solution contains a Critical Error and is therefore invalid" / "The solution's approach is viable but contains Justification Gaps."
- **List of Findings:** For each issue:
  - **Location:** Quote the key phrase or equation.
  - **Issue:** Critical Error (breaks logical chain: fallacies, factual errors) or Justification Gap (conclusion may hold but argument incomplete).

### Step 3: Review Verification Report

When the proof is difficult or fails verification, consider whether a counterexample exists or the proposition is false before choosing outcomes (3) or (4). Decide which of the following applies:

| Outcome | Action |
|---------|--------|
| **(1) Proof complete, acceptable** | Present the proof to the user. Done. |
| **(2) Proof has errors, needs correction** | When correcting, provide to the LLM: **original problem**, **current problematic proof**, and **verification report**. Generate a corrected proof, then return to Step 2. |
| **(3) Proof cannot be completed; need user hints** | Present the current proof and verification report to the user. Ask for hints. After the user responds, regenerate the proof and return to Step 1. |
| **(4) Proof cannot be completed; need to modify definition or proposition** | Present the current proof and verification report to the user. Suggest modifications to definitions or the proposition itself. Wait for user input, then regenerate and return to Step 1. |
| **(5) Proposition is false** | State that the proposition is false. Give reasons or a counterexample. Done. |

## Verification Checklist

Before declaring a proof complete:

- [ ] Every implication and equality is justified
- [ ] No logical fallacies (e.g., subtracting inequalities without care)
- [ ] No unjustified limit/order interchanges (e.g., limit and integral)
- [ ] Case splits are exhaustive and each case is handled
- [ ] All key lemmas are stated precisely and proven or cited
- [ ] Notation is consistent; definitions match usage

## Example Structure

**Summary**

- **a. Verdict:** I have successfully proved the proposition. The set $S$ is finite if and only if...
- **b. Method Sketch:** We first reduce to the case where $X$ is compact. The key lemma is: [precise statement]. We then apply induction on the dimension.

**Detailed Solution**

[Full proof with numbered steps, each justified.]