## Code Comparison: LLM vs. LLM + APOLLO ### Overview The image presents a side-by-side comparison of code snippets, labeled "LLM" on the left and "LLM + APOLLO" on the right. The code appears to be related to mathematical theorem proving, likely using a tool like Lean. The comparison highlights differences in the code required to achieve the same result with and without the APOLLO extension. Specific lines are highlighted in red (for LLM) and green (for LLM + APOLLO) to indicate differences. ### Components/Axes * **Titles:** "LLM" (top-left, red background) and "LLM + APOLLO" (top-right, green background). * **Code:** The main content consists of code blocks written in a formal language, likely a theorem prover's syntax. * **Highlights:** Specific lines of code are highlighted with colored backgrounds (red and green) and numbered annotations (#1, #2, #3). ### Detailed Analysis or Content Details **Left Side (LLM):** * Imports: * `import Mathlib` * `import Aesop` * Option Setting: * `set_option maxHeartbeats 0` * Open Statement: * `open BigOperators Real Nat Topology Rat` * Theorem Definition: * `theorem mathd_algebra_158_llm (a k : N) (h₀ : Even a)` * `(h₁ : ↑(Σ k in Finset.range 8, 2 * k + 1) - ↑(Σ k in Finset.range 5, a + 2 * k) = (4 : Z)) :` * `a = 8 := by` * `simp_all` * `ring_nf at *` * `have eq : (5 : Z) * (↑a : Z) + (2 : Z) * (↑k : Z) = 53 := by` * `linarith` * Highlighted Sections: * **#1 (Red):** `have h_even : ∃ (m : N), a = 2 * m := h₀` * **#2 (Red):** `have two_divides_53 : ∃ (d : Z), 53 = 2 * d := by` * `rw [← even_sum] at eq_subst` * `rw [eq_subst] at eq` * `exact <(↑5) * (↑m : Z) + ↑k, eq>` * `have not_divides : ¬ ∃ (d : Z), 53 = 2 * d := by` * `rintro <d, hd>` * `have mod53 : (53 : Z) % 2 = 1 := by` * `norm_num` * `have mod_d : (2 * d) % 2 = 0 := by` * `norm_num` * `rw [hd] at mod53` * **#3 (Red):** `rw [mul_mod] at mod53` * `linarith` * `have contra : False := by` * `apply not_divides` * `exact two_divides_53` * `exact False.elim contra` **Right Side (LLM + APOLLO):** * Imports: * `import Mathlib` * `import Aesop` * Option Setting: * `set_option maxHeartbeats 0` * Open Statement: * `open BigOperators Real Nat Topology Rat` * Theorem Definition: * `theorem mathd_algebra_158_apollo (a k : N) (h₀ : Even a)` * `(h₁ : ↑(Σ k in Finset.range 8, 2 * k + 1) - ↑(Σ k in Finset.range 5, a + 2 * k) = (4 : Z)) :` * `a = 8 := by` * `simp_all` * `ring_nf at *` * `have eq : (5 : Z) * (↑a : Z) + (2 : Z) * (↑k : Z) = 53 := by` * `linarith` * Highlighted Sections: * **#1 (Green):** `have h_even : ∃ (m : N), a = 2 * m := by` * `exact Even.exists_two_nsmul a h₀` * **#2 (Green):** `have two_divides_53 : ∃ (d : Z), 53 = 2 * d := by` * `:= by omega` * `have not_divides : ¬ ∃ (d : Z), 53 = 2 * d := by` * `rintro <d, hd>` * `have mod53 : (53 : Z) % 2 = 1 := by` * `norm_num` * `have mod_d : (2 * d) % 2 = 0 := by` * `norm_num` * `rw [hd] at mod53` * **#3 (Green):** `rw [mul_mod] at mod53` * `linarith` * `have contra : False := by` * `apply not_divides` * `exact two_divides_53` * `exact False.elim contra` ### Key Observations * The code on both sides aims to prove the same theorem (`mathd_algebra_158`). * The "LLM + APOLLO" version appears to be more concise in certain sections, particularly in the highlighted areas. * The highlighted sections (#1, #2, #3) show where APOLLO simplifies the proof process. * In #1, APOLLO provides a direct proof (`exact Even.exists_two_nsmul a h₀`) compared to the original. * In #2, APOLLO uses `:= by omega` to automatically prove the statement. * The rest of the code is identical. ### Interpretation The image demonstrates the benefits of using the APOLLO extension in a theorem proving environment. APOLLO seems to automate or simplify certain proof steps, leading to more concise and potentially more efficient code. The highlighted sections illustrate specific instances where APOLLO reduces the amount of manual reasoning required. The comparison suggests that APOLLO can improve the user experience and potentially reduce the complexity of theorem proving tasks.