## Diagram: Mathematical Proof/Code Snippets ### Overview The image presents a side-by-side comparison of mathematical code and proof snippets, likely generated by a Large Language Model (LLM) and a system called "APOLLO". The content appears to be related to real number topology and calculations involving square roots and norms. The diagram is structured as a series of code blocks with annotations indicating the flow of logic or derivation steps. The left side is labeled "LLM", the center "APOLLO", and the right side "LLM + APOLLO". ### Components/Axes The diagram is not a traditional chart with axes. Instead, it consists of several distinct code blocks and text annotations. Key components include: * **Headers:** "LLM", "APOLLO", "LLM + APOLLO" at the top of each column. * **Code Blocks:** Representing mathematical expressions and code snippets. These are formatted with indentation and annotations. * **Annotations:** Textual notes (e.g., "by positivity", "by norm num", "calc") indicating the reasoning or operation applied. * **Mathematical Expressions:** Involving `Real.sqrt`, `Real.mul`, `Real.Topology Rat`, `norm`, `norm num`, and variables like `x`, `y`, `z`. * **Footer:** Contains information about the model versions and a timestamp. ### Detailed Analysis or Content Details **LLM Column (Left):** * `import Mathlib` * `import Aesop` * `set_option maxHeartbeats 0` * `open BigOperators Real Nat Topology Rat` * `theorem mathd_algebra_293_llm (x : NNReal) : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (36 * x) * Real.sqrt (35 * x) := by` * `have hi : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * x) * (12 * x) * Real.sqrt (63 * x) := by` * `rw [Real.sqrt_mul (by positivity), Real.sqrt_mul (by positivity)]` * `have h1 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `rw [Real.sqrt_mul (by positivity), Real.sqrt_mul (by positivity)]` * `have h2 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `simp` * `trace_state 'show [norm x, try rfl by positivity]'` * `apply h1; try ring_nf; abel` * `have h3 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `rw [Real.sqrt_mul (by positivity), Real.sqrt_mul (by positivity)]` * `have h4 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `simp` * `trace_state 'show [norm x, try rfl by positivity]'` * `apply h3; try ring_nf; abel` **APOLLO Column (Center):** * `import Mathlib` * `import Aesop` * `set_option maxHeartbeats 0` * `open BigOperators Real Nat Topology Rat` * `theorem mathd_algebra_293_apollo (x : NNReal) : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (36 * x) * Real.sqrt (35 * x) := by` * `have hi : Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) = Real.sqrt (60 * x) * (12 * x) * Real.sqrt (63 * x) := by` * `rw [Real.sqrt_mul (by positivity)]` * `have h1 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `rw [Real.sqrt_mul (by positivity)]` * `have h2 : (60 * x) * (12 * x) * (63 * x) = (36 * x) * (35 * x) := by` * `calc` * `√(15 * 3 * 7) := by` * `√(15 * 3 * 7) := by` * `rw [Real.sqrt_mul (by norm num)]` * `have h2a : √(315) = √(3 * 3 * 5 * 7) := by` * `(36 * 12) + √(21) := √(3 * √(3 * √(5 * 7))) := by` * `have h2b : √(315) = √(35 * 9) := by` * `calc` * `√(35 * 9) = 3 * √(35) := by` * `rw [Real.sqrt_mul (by norm num)]` * `have h2b1 : √(35 * 9) = 3 * √(35) := by` * `rw [Real.sqrt_mul (by norm num)]` **LLM + APOLLO Column (Right):** * `have h2 : (√ (15) * √ (3) * √ (7)) * √ (7) := by` * `have h2 : (√ (15) * √ (3) * √ (7)) * √ (7) := by` * `calc` * `√(15 * 3 * 7) := by` * `√(15 * 3 * 7) := by` * `rw [Real.sqrt_mul (by norm num)]` * `have h2a : √(315) = √(3 * 3 * 5 * 7) := by` * `(36 * 12) + √(21) := √(3 * √(3 * √(5 * 7))) := by` * `have h2b : √(315) = √(35 * 9) := by` * `calc` * `√(35 * 9) = 3 * √(35) := by` * `rw [Real.sqrt_mul (by norm num)]` * `have h2b1 : √(35 * 9) = 3 * √(35) := by` * `rw [Real.sqrt_mul (by norm num)]` **Footer:** * `model_version = 'v0.3.26'` * `date = '2024-01-26'` * `commit = '8f8f599'` ### Key Observations * The LLM and APOLLO columns attempt to prove the same theorem. * The APOLLO column includes more detailed calculation steps ("calc") and intermediate results. * Both columns utilize `Real.sqrt_mul` with the justification "by positivity" or "by norm num". * The "trace\_state" annotations in the LLM column suggest debugging or tracing of the proof process. * The LLM + APOLLO column appears to be a combination of the outputs from the other two columns. ### Interpretation The diagram demonstrates a comparison of mathematical reasoning and proof generation between a standard LLM and the APOLLO system. APOLLO seems to provide a more step-by-step, detailed breakdown of the calculation, potentially making the proof more transparent and easier to verify. The use of "calc" and intermediate results suggests a more explicit computational approach. The LLM, while attempting the same proof, relies more on higher-level transformations and annotations like "trace\_state". The combination in the "LLM + APOLLO" column suggests a potential synergy where APOLLO's detailed calculations can augment the LLM's reasoning capabilities. The footer information indicates the specific versions of the models and the date of the output, providing context for reproducibility and analysis. The overall goal appears to be to evaluate and improve the ability of LLMs to perform rigorous mathematical proofs. The repeated use of `Real.sqrt_mul` suggests this is a core operation in the proof process, and the "by positivity" justification indicates a reliance on the non-negativity of real numbers.