mathematical-proofs-mentor skill

Overview

This skill is a Mathematical Proofs Mentor that guides students through proof techniques, rigorous argumentation, and structured reasoning. It focuses on helping learners choose strategies, spot logical gaps, and improve clarity in written proofs. Note: the material originates from a legacy template and requires research validation before being treated as fully authoritative.

How this skill works

The mentor inspects submitted proof drafts, identifies missing steps, uncovers implicit assumptions, and suggests tighter formulations or alternative strategies (direct, contrapositive, contradiction, induction, combinatorial, etc.). It provides targeted prompts to help the student fill gaps rather than handing over complete solutions, and can rewrite arguments for clarity and concision. When confidence is low for a specific topic, it flags areas that need up-to-date references or further verification.

When to use it

• When drafting or revising homework proofs and seeking feedback on logical structure.
• While learning new proof techniques and wanting guided practice on strategy selection.
• To check proofs for hidden assumptions, unclear quantifier use, or unjustified steps.
• When preparing solutions for presentations or write-ups that require rigorous exposition.
• If you want help converting informal arguments into formal, step-by-step proofs.

Best practices

• Provide the full problem statement, definitions, and any hypotheses up front.
• Include your current attempt and mark steps you’re unsure about.
• State which proof styles you prefer or want to practice (e.g., induction, contradiction).
• Ask for hints or checkpoints rather than full solutions to maximize learning.
• Request rewritten versions at different rigor levels: sketch, standard, or formal.

Example use cases

• A student submits a draft induction proof and gets pinpointed base case errors and a corrected induction hypothesis.
• Converting an informal combinatorial argument into a clear bijective proof with explicit mapping.
• Identifying and removing hidden uses of excluded middle in constructive proofs.
• Polishing a topology proof to ensure epsilon-delta or neighborhood arguments are fully explicit.
• Preparing lecture-ready proofs by tightening language, quantifier order, and logical flow.

FAQ