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:
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.]
1---2name: prove-math-proposition3description: Prove Research-Level Mathematical Propositions4---56# Prove Research-Level Mathematical Propositions78Three-step workflow: generate proof → verify (report) → review report → accept, correct, request hints, modify proposition, or identify falsehood.910## Core Principles1112- **Rigor is paramount.** Every step must be logically sound and clearly explained. A correct conclusion from flawed reasoning is a failure.13- **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).14- **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.15- **Use TeX** for all mathematics: `$...$` for inline, `$$...$$` for display.1617## Output Format1819Structure the response into these sections, in order:2021### 1. Summary2223**a. Verdict**24- Complete: "I have successfully proved the proposition. The conclusion is..."25- Partial: "I have not found a complete proof, but I have rigorously proved that..."2627**b. Method Sketch**28- High-level strategy and logical flow29- Full, precise statements of key lemmas or major intermediate results30- Key constructions or case splits, if applicable3132### 2. Detailed Solution3334Full 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.3536## Workflow3738### Step 1: Generate Proof3940Produce a proof following the Output Format above (Summary + Detailed Solution).4142### Step 2: Verify Proof4344Act as an IMO grader. Produce a **Verification Report**:4546- **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."47- **List of Findings:** For each issue:48 - **Location:** Quote the key phrase or equation.49 - **Issue:** Critical Error (breaks logical chain: fallacies, factual errors) or Justification Gap (conclusion may hold but argument incomplete).5051### Step 3: Review Verification Report5253When 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:5455| Outcome | Action |56|---------|--------|57| **(1) Proof complete, acceptable** | Present the proof to the user. Done. |58| **(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. |59| **(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. |60| **(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. |61| **(5) Proposition is false** | State that the proposition is false. Give reasons or a counterexample. Done. |6263## Verification Checklist6465Before declaring a proof complete:6667- [ ] Every implication and equality is justified68- [ ] No logical fallacies (e.g., subtracting inequalities without care)69- [ ] No unjustified limit/order interchanges (e.g., limit and integral)70- [ ] Case splits are exhaustive and each case is handled71- [ ] All key lemmas are stated precisely and proven or cited72- [ ] Notation is consistent; definitions match usage7374## Example Structure7576**Summary**7778- **a. Verdict:** I have successfully proved the proposition. The set $S$ is finite if and only if...79- **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.8081**Detailed Solution**8283[Full proof with numbered steps, each justified.]
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