zaoqu-liu Theorem Proving Skill
Description
Assist with constructing, verifying, and typesetting mathematical proofs. Combines rigorous logical reasoning with computational verification.
Tools Used
latex_compile- Typeset proofs and mathematical documents (auto-switches to LaTeX editor)update_latex- Write LaTeX content to the editor for review before compilingjupyter_execute- Verify results computationally (sympy, numpy)update_notes- Write proof outlines and scratch work to Notes editor
Capabilities
Proof Construction
- Direct proofs, proof by contradiction, proof by induction
- Constructive and non-constructive existence proofs
- Epsilon-delta arguments in analysis
- Diagram chasing in algebra/category theory
Verification
- Symbolic computation to check algebraic manipulations
- Numerical examples to build intuition
- Counterexample search for false conjectures
- Automated checking of special cases
Typesetting
- AMS theorem environments (theorem, lemma, proposition, corollary, definition)
- Proper mathematical notation and spacing
- Cross-references and equation numbering
- Multi-part proofs with clear structure
Usage Patterns
Prove a Theorem
When user says: "Prove that [statement]"
- Clarify definitions and assumptions
- Outline proof strategy
- Construct formal proof step-by-step
- Verify key steps computationally if possible
- Typeset in LaTeX with proper environments
Verify a Conjecture
When user says: "Is it true that [conjecture]?"
- Test with specific examples (jupyter_execute)
- Search for counterexamples
- Attempt proof if examples support it
- Report findings with confidence level