theorem-proving

# 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 compiling - `jupyter_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]" 1. Clarify definitions and assumptions 2. Outline proof strategy 3. Construct formal proof step-by-step 4. Verify key steps computationally if possible 5. Typeset in LaTeX with proper environments ### Verify a Conjecture When user says: "Is it true that [conjecture]?" 1. Test with specific examples (jupyter_execute) 2. Search for counterexamples 3. Attempt proof if examples support it 4. Report findings with confidence level ## Tool Examples ### Typeset a theorem in LaTeX ``` update_latex content="\\documentclass{article}\n\\usepackage{amsthm,amsmath}\n\\newtheorem{theorem}{Theorem}\n\\begin{document}\n\\begin{theorem}\nFor all $n \\geq 1$, $\\sum_{k=1}^{n} k = \\frac{n(n+1)}{2}$.\n\\end{theorem}\n\\begin{proof}\nBy induction on $n$. Base case $n=1$: $1 = \\frac{1 \\cdot 2}{2}$. Inductive step: assume true for $n$, then $\\sum_{k=1}^{n+1} k = \\frac{n(n+1)}{2} + (n+1) = \\frac{(n+1)(n+2)}{2}$.\n\\end{proof}\n\\end{document}" ``` ### Verify computationally with SymPy ``` jupyter_execute code="from sympy import symbols, summation, simplify\nk, n = symbols('k n', positive=True, integer=True)\nresult = simplify(summation(k, (k, 1, n)) - n*(n+1)/2)\nprint(f'Difference: {result}') # Should be 0" ```