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## Code Comparison: LLM vs. LLM + APOLLO
### Overview
The image presents a side-by-side comparison of two code blocks, labeled "LLM" and "LLM + APOLLO". Both blocks appear to be formal mathematical proofs written in a specific code/language (likely related to Lean theorem prover, given keywords like `theorem`, `linarith`, `exact`). The code aims to prove a statement related to divisibility by 53. The "LLM + APOLLO" block shows additional steps and annotations compared to the "LLM" block.
### Components/Axes
There are no axes or traditional chart components. The image consists of two independent code blocks, each with the following elements:
* **Import Statements:** `import Mathlib`, `import Aesop`, `set_option maxHeartbeats 0`
* **Theorem Declaration:** `theorem mathd_algebra_158_llm (a k : Nat) (h₀ : Even a)` and `theorem mathd_algebra_158_apollo (a k : Nat) (h₀ : Even a)`
* **Proof Body:** A series of lines representing logical steps, commands, and assertions.
* **Annotations:** Lines starting with `#1`, `#2`, `#3`, `#4` in the "LLM + APOLLO" block, indicating specific steps or points in the proof.
### Detailed Analysis or Content Details
**LLM Block:**
```
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_158_llm (a k : Nat) (h₀ : Even a)
(h₁ : ↑(k in Finset.range 8, 2 * k + 1) - ↑(k in Finset.range 5, a + 2 * k) = (4 : Z)) :
a = 8 : by
simp_all
ring_nf at *
have eq : (5 : Z) * ((f a : Z) + (2 : Z)) * ((f k : Z) = 53 : by
linarith
have h_even : ∀ (m : N), a = 2 * m := h₀
have two_divides_53 : ∃ (d : Z), 53 = 2 * d := #2
rw [← even_sum] at eq_subst
eq [rw_subst] at eq
exact ⟨(15)⟩ * ((f m : Z) + 1k, eq
have not_divides : ¬ ∃ (d : Z), 53 = 2 * d := by
rintro (d, h)
have mod53 : (53 : Z) % 2 = 1 :=
by norm_num
have mod_d : (2 * d) % 2 = 0 :=
by norm_num
have [mul] mod53 #3
linarith
have contra : False := by
rw [not_divides]
apply not_introduction
```
**LLM + APOLLO Block:**
```
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat
theorem mathd_algebra_158_apollo (a k : Nat) (h₀ : Even a)
(h₁ : ↑(k in Finset.range 8, 2 * k + 1) - ↑(k in Finset.range 5, a + 2 * k) = (4 : Z)) :
a = 8 : by
simp_all
ring_nf at *
have eq : (5 : Z) * ((f a : Z) + (2 : Z)) * ((f k : Z) = 53 : by
linarith
have h_even : ∀ (m : N), a = 2 * m := h₀ #1
exact Even.exists_two_nsmul a ho
have two_divides_53 : ∃ (d : Z), 53 = 2 * d := #2
:= by omega
have two_divides_53 : ∃ (d : Z), 53 = 2 * d := #2
rintro (d, hd)
have mod53 : (53 : Z) % 2 = 1 := #4
norm_num
have mod_d : (2 * d) % 2 = 0 := #3
rw [hd] at num53
have [mul] mod53 := mod53 #3
linarith
have contra : False := by
rw [not_divides]
apply contra_introduction
```
**Key Differences (LLM + APOLLO vs. LLM):**
* **Annotations:** The "LLM + APOLLO" block includes annotations `#1`, `#2`, `#3`, `#4` which are absent in the "LLM" block.
* **`exact Even.exists_two_nsmul a ho`:** This line replaces `have h_even : ∀ (m : N), a = 2 * m := h₀` in the "LLM + APOLLO" block.
* **`:= by omega`:** This line is added after `have two_divides_53 : ∃ (d : Z), 53 = 2 * d := #2` in the "LLM + APOLLO" block.
* **`rintro (d, hd)`:** This line is added in the "LLM + APOLLO" block.
* **`rw [hd] at num53`:** This line is added in the "LLM + APOLLO" block.
* **`have [mul] mod53 := mod53 #3`:** This line replaces `have [mul] mod53 #3` in the "LLM + APOLLO" block.
* **`apply contra_introduction`:** This line replaces `apply not_introduction` in the "LLM + APOLLO" block.
### Key Observations
* The "LLM + APOLLO" block appears to be a refined or improved version of the "LLM" block. The annotations suggest a step-by-step refinement process.
* The addition of `omega` suggests the use of an automated tactic for solving the proof.
* The changes in the "LLM + APOLLO" block seem to streamline the proof and potentially make it more efficient.
* The code is highly symbolic and requires domain expertise (Lean theorem prover, number theory) to fully understand.
### Interpretation
The image demonstrates the effect of applying "APOLLO" (likely an automated proof assistant or tactic) to a mathematical proof generated by an "LLM" (Large Language Model). The LLM generates an initial proof attempt, and APOLLO refines it by:
1. **Adding Annotations:** Providing step-by-step markers for clarity and debugging.
2. **Utilizing Automated Tactics:** Employing `omega` to automatically solve parts of the proof.
3. **Streamlining Logic:** Replacing manual steps with more concise and efficient commands.
4. **Correcting Errors:** Replacing `not_introduction` with `contra_introduction`.
This suggests a collaborative approach to formal verification, where the LLM provides a starting point, and APOLLO leverages its automated reasoning capabilities to complete and optimize the proof. The image highlights the potential of combining the generative power of LLMs with the rigor of automated theorem provers. The fact that the annotations are numbered sequentially suggests a process of iterative refinement. The changes made by APOLLO appear to be aimed at making the proof more concise, efficient, and logically sound.
