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"