Theorem Proving Skill

---
name: theorem-proving-d4dda77f4390
description: "Construct and verify mathematical proofs using LaTeX typesetting and computational verification via jupyter_execute. Use when the user asks to prove a theorem, verify a mathematical argument, construct a formal proof, or check proof correctness computationally."
---

# 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"
```