## Code Comparison: LLM vs LLM + APOLLO
### Overview
The image shows two Coq code snippets (a formal proof assistant language) comparing implementations of a mathematical theorem under two systems: "LLM" (red header) and "LLM + APOLLO" (green header). Both snippets define a theorem about real number square root properties and provide partial proofs.
### Components/Axes
- **Headers**:
- Red header: "LLM" (left code block)
- Green header: "LLM + APOLLO" (right code block)
- **Imports**:
- Both import `Mathlib` and `Aesop` libraries.
- **Options**:
- `set_option maxHeartbeats 0` (disables interactive proof steps).
- **Theorems**:
- Left: `mathd_algebra_293_llm`
- Right: `mathd_algebra_293_apollo`
- **Proof Steps**:
- Both use `rw` (rewrite) and `ring_nf` (normalize ring expressions).
- Right code adds `mul_nonneg` and `exact mul_nonneg` steps.
### Detailed Analysis
#### Left Code (LLM)
1. **Theorem Statement**:
```coq
theorem mathd_algebra_293_llm (x : NNRReal) :
Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) :=
36 * x * Real.sqrt (35 * x)
```
2. **Proof Steps**:
- Uses `rw [← Real.sqrt_mul (by positivity)]` to rewrite square root products.
- Applies `ring_nf` to simplify ring expressions.
- Ends with `ring_nf` and a highlighted `nlinarith` (nonlinear arithmetic) tactic.
#### Right Code (LLM + APOLLO)
1. **Theorem Statement**:
```coq
theorem mathd_algebra_293_apollo (x : NNRReal) :
Real.sqrt (60 * x) * Real.sqrt (12 * x) * Real.sqrt (63 * x) :=
36 * x * Real.sqrt (35 * x)
```
2. **Proof Steps**:
- Adds `mul_nonneg` and `exact mul_nonneg` to handle non-negative multiplication.
- Uses `mul_nonneg_iff_of_pos_right` to link non-negativity with multiplication.
- Includes `norm_num` and `have` clauses to enforce constraints (e.g., `0 ≤ x`).
- Ends with `exact mul_nonneg (mul_nonneg A B) C` to finalize the proof.
### Key Observations
1. **Structural Similarity**:
- Both theorems share identical mathematical statements but differ in proof strategies.
2. **APOLLO Enhancements**:
- The right code explicitly handles non-negativity (`mul_nonneg`) and uses more granular lemmas (e.g., `mul_nonneg_iff_of_pos_right`).
- The left code relies on `nlinarith`, while the right code uses `exact` to finalize proofs.
3. **Complexity**:
- The right code’s proof is longer, suggesting APOLLO provides more foundational support for edge cases.
### Interpretation
The APOLLO extension enhances the LLM system by:
- **Rigorous Non-Negativity Handling**: Explicitly enforcing `0 ≤ x` via `mul_nonneg` to avoid invalid square root operations.
- **Modular Proof Construction**: Breaking proofs into smaller, verifiable steps (e.g., `have A : 0 ≤ x`) for clarity and correctness.
- **Advanced Lemmas**: Leveraging `mul_nonneg_iff_of_pos_right` to connect algebraic properties with topological constraints.
The differences highlight APOLLO’s role in refining formal proofs, ensuring robustness in mathematical reasoning. The left code’s brevity may sacrifice explicitness for conciseness, while the right code prioritizes correctness through detailed case analysis.
