## Code Comparison: LLM vs. LLM + APOLLO
### Overview
The image presents a side-by-side comparison of code snippets, one labeled "LLM" and the other "LLM + APOLLO". Both snippets appear to be related to mathematical theorem proving, likely using a formal verification system. The comparison highlights differences in the code required to prove a specific theorem using two different approaches.
### Components/Axes
* **Left Block:**
* Title: "LLM" (red background)
* Code: A series of commands and a theorem definition.
* **Right Block:**
* Title: "LLM + APOLLO" (green background)
* Code: A series of commands and a theorem definition.
### Detailed Analysis or ### Content Details
**Left Block (LLM):**
* `import Mathlib`
* `import Aesop`
* `set_option maxHeartbeats 0`
* `open BigOperators Real Nat Topology Rat`
* `theorem imo_1983_p6_llm (a b c : R) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) (h₁ : c < a + b) (h₂ : b < a + c) (h₃ : a < b + c) : 0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by`
* `nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a), sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a), mul_pos h₀.left h₀.right.left, mul_pos h₀.left h₀.right.right, mul_pos h₀.right.left h₀.right.right]`
* Annotation: "#1" next to `sq_nonneg (b - c)` in the `nlinarith` line.
**Right Block (LLM + APOLLO):**
* `import Mathlib`
* `import Aesop`
* `set_option maxHeartbeats 0`
* `open BigOperators Real Nat Topology Rat`
* `theorem imo_1983_p6_apollo (a b c : R) (h₀ : 0 < a ∧ 0 < b ∧ 0 < c) (h₁ : c < a + b) (h₂ : b < a + c) (h₃ : a < b + c) : 0 ≤ a ^ 2 * b * (a - b) + b ^ 2 * c * (b - c) + c ^ 2 * a * (c - a) := by`
* `nlinarith [sq_nonneg (a - b), sq_nonneg (b - c), sq_nonneg (c - a), mul_pos (sub_pos.mpr h₁), (sub_pos.mpr h₂), mul_pos (sub_pos.mpr h₂), (sub_pos.mpr h₃), mul_pos (sub_pos.mpr h₃ (sub_pos.mpr h₁)]`
* Annotation: "#1" next to `sq_nonneg (b - c)` in the `nlinarith` line.
### Key Observations
* Both code snippets define a theorem named `imo_1983_p6_llm` and `imo_1983_p6_apollo` respectively, which appears to be related to the International Mathematical Olympiad (IMO) problem from 1983, problem 6.
* The theorem states that under certain conditions on real numbers `a`, `b`, and `c`, a specific inequality holds.
* The `nlinarith` command is used to prove the theorem. This command likely invokes a linear arithmetic solver.
* The key difference lies in the arguments passed to the `nlinarith` command. The "LLM" version requires more explicit hints (multiple repetitions of `sq_nonneg` and explicit calls to `mul_pos` with detailed arguments), while the "LLM + APOLLO" version uses a more concise and automated approach (using `sub_pos.mpr` and fewer explicit hints).
### Interpretation
The image demonstrates the impact of using the "APOLLO" extension in conjunction with "LLM" for proving mathematical theorems. The "LLM + APOLLO" approach significantly simplifies the proof process by reducing the amount of manual guidance required for the linear arithmetic solver. This suggests that "APOLLO" provides a higher level of automation or more powerful reasoning capabilities compared to using "LLM" alone. The annotation "#1" likely refers to a specific point or step in the proof process that is being highlighted.
