---
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"
```


---
Source attribution: Prismer-AI
Original project: https://github.com/Prismer-AI/Prismer
License: MIT
License information: https://opensource.org/license/mit
Original upstream instructions; downloaded from Mega AI Hub.

## Upstream license notice

MIT License

Copyright (c) 2026 Gabriel Moreira

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all
copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.



# Mega AI Hub resource metadata

Creator / publisher: Prismer-AI
Original project: https://github.com/Prismer-AI/Prismer
Attribution basis: Upstream repository explicitly recorded in package mirror path

Fields of work: Education & Research
Roles: Researcher, Tutor / Instructor, Academic Writer
Specialties: API
Tools: Tool not specified
Country applicability: Country not specified
Classification basis: Source category, role and specialization labels
Instruction SHA-256: 997199fcb17a6039541260e27e661b4f2656a85774177b6f8857579cb00106a0