## Code Comparison: LLM vs. LLM + APOLLO ### Overview The image presents a side-by-side comparison of code snippets, labeled "LLM" (left, red background) and "LLM + APOLLO" (right, green background). The code appears to be a formal proof or derivation, possibly in a language like Lean or Coq, dealing with number theory and algebraic manipulations. The comparison highlights differences in the proof strategies or steps taken by the two approaches. ### Components/Axes * **Headers:** * Left (Red): "LLM" * Right (Green): "LLM + APOLLO" * **Code Blocks:** Two columns of code, each representing a different approach to the same theorem. * **Annotations:** Numerical annotations (e.g., "#1", "#2") are present within the code, possibly indicating specific points of interest or differences. ### Detailed Analysis or ### Content Details **Left Column (LLM):** * **Theorem Statement:** `theorem mathd_algebra_289_1lm (k t m n: N) (h0: Nat.Prime m ∧ Nat.Prime n) (h1: t < k) (h2: (k ^ 2: Z) - m * k + n = 0) (h3: (t ^ 2: Z) - m * t + n = 0): m ^ n + n ^ m + k * t + t ^ k = 20 := by` * **Key Steps:** * Derives `have h4: (k : Z) ^ 2 - m * k + n = 0 := by exact_mod_cast h2` * Derives `have h5: (t : Z) ^ 2 - m * t + n = 0 := by exact_mod_cast h3` * Uses `linarith` to derive several intermediate results. * Applies `mul_eq_zero.mp h9` and `resolve_left` with `omega`. * Uses `nlinarith` with `[h4, h5, h12]`. * Derives `have k = 1 := by nlinarith #1` * Derives `have t = 1 := by omega` * Derives `have n = 1 := by omega #2` * Derives `have k = 1 := by nlinarith #3` * Uses `interval_cases k <;> omega #4` **Right Column (LLM + APOLLO):** * **Theorem Statement:** `theorem mathd_algebra_289_apollo (k t m n: N) (h0: Nat.Prime m ∧ Nat.Prime n) (h1: t < k) (h2: (k ^ 2: Z) - m * k + n = 0) (h3: (t ^ 2: Z) - m * t + n = 0): m ^ n + n ^ m + k * t + t ^ k = 20 := by` * **Key Steps:** * Similar initial steps as LLM, deriving `h4` and `h5` using `exact_mod_cast`. * Derives `have h24: (k : Z) + (t : Z) = m := by #5 exact h12` * Uses `gcongr` and `rw` (rewrite) at various points. * Applies `simp_all` with various arguments like `Nat.cast_add`, `Nat.cast_one`, etc. * Uses `by_contra h` and `push_neg at h` for proof by contradiction. * Derives `have k % 2 = 0 := by omega #6` * Derives `have (m : Z) = (k : Z) + 1 := by linarith #7` * Derives `have k = 2 := by interval_cases k <;> omega #8` **Annotations:** * `#1`, `#2`, `#3`, `#4`, `#5`, `#6`, `#7`, `#8`: These annotations highlight specific lines or sections of code, likely indicating points of comparison or significant steps in the proof. ### Key Observations * Both approaches start with the same theorem and initial assumptions. * The "LLM + APOLLO" approach seems to incorporate more explicit simplifications and rewrites using lemmas and theorems from the `Nat` library. * The use of `interval_cases` and proof by contradiction (`by_contra h`) is more prominent in the "LLM + APOLLO" approach. * The annotations suggest specific points where the two approaches diverge or where key steps are taken. ### Interpretation The image demonstrates two different strategies for proving the same theorem. The "LLM + APOLLO" approach likely benefits from additional knowledge or tools (represented by "APOLLO") that allow for more direct simplification and application of relevant lemmas. The annotations highlight the critical junctures where these differences in strategy become apparent. The presence of "omega" suggests the use of an automated theorem prover to discharge certain subgoals. The comparison suggests that "LLM + APOLLO" may be more efficient or require less manual intervention than the pure "LLM" approach.

## Code Comparison Diagram: LLM vs. LLM + APOLLO (Lean 4 Proofs) ### Overview This image presents a side-by-side comparison of two formal mathematical proofs written in the Lean 4 theorem proving language. The image is divided into two main panels. The left panel, outlined in red and titled "LLM", shows a relatively short, single-column proof containing several highlighted errors or hallucinated steps. The right panel, outlined in green and titled "LLM + APOLLO", shows a much longer, three-column proof where the errors from the left panel have been replaced with extensive, successfully verified tactic blocks. The numbered tags (#1 through #8) map the failed steps in the standalone LLM to the corrected, expanded sub-proofs generated by the APOLLO system. *Language Declaration:* The text is written in Lean 4, a formal programming and theorem-proving language that utilizes English keywords alongside mathematical notation. It is transcribed exactly as written. ### Components/Layout * **Left Panel (Red Bounding Box):** * **Header:** "LLM" (white text on red background). * **Layout:** A single column of code. * **Highlights:** Specific lines of code are highlighted in light red. These lines contain inline comments with numbers (e.g., `#1`, `#2`), indicating points of failure or hallucination where the LLM assumed a tactic (like `omega` or `nlinarith`) would solve the goal, but it did not. * **Right Panel (Green Bounding Box):** * **Header:** "LLM + APOLLO" (white text on green background). * **Layout:** Three columns of code, separated by thin vertical gray lines. The second and third columns have `...` at the top, indicating continuous flow from the previous column. * **Highlights:** Large blocks of code are highlighted in light green. These blocks contain corresponding inline comments (`#1`, `#2`, etc.). These represent the expanded, rigorous proof steps that APOLLO generated to successfully bridge the logical gaps left by the standalone LLM. --- ### Content Details (Transcription) #### Left Panel: LLM (Red) ```lean theorem mathd_algebra_289_llm (k t m n : ℕ) (h₀ : Nat.Prime m ∧ Nat.Prime n) (h₁ : t < k) (h₂ : (k ^ 2 : ℤ) - m * k + n = 0) (h₃ : (t ^ 2 : ℤ) - m * t + n = 0) : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by have h4 : (k : ℤ) ^ 2 - m * k + n = 0 := by exact_mod_cast h₂ have h5 : (t : ℤ) ^ 2 - m * t + n = 0 := by exact_mod_cast h₃ have h6 : (k : ℤ) ^ 2 - m * k + n - ((t : ℤ) ^ 2 - m * t + n) = 0 := by linarith have h7 : (k : ℤ) ^ 2 - (t : ℤ) ^ 2 - m * (k - t) = 0 := by linarith have h8 : (k - t) * ((k : ℤ) + (t : ℤ)) - m * (k - t) = 0 := by ring_nf at h7 ⊢ linarith have h9 : (k - t) * ((k : ℤ) + (t : ℤ) - m) = 0 := by linarith have h10 : (k : ℤ) + (t : ℤ) - m = 0 := by apply (mul_eq_zero.mp h9).resolve_left omega have h11 : (k : ℤ) + (t : ℤ) = m := by linarith have h12 : k + t = m := by norm_num at h11 ⊢ exact_mod_cast h11 have h14 : k * t = n := by nlinarith [h4, h5, h12] have h15 : t = 1 := by have h16 : t = 1 := by have h17 : k * t = n := h14 have h18 : t < k := h₁ have h19 : k > 1 := by by_contra h push_neg at h have : k = 1 := by omega [RED HIGHLIGHT START] have t : 1 := by nlinarith #1 [RED HIGHLIGHT END] linarith have h20 : t = 1 := by have h21 : k * t = n := h17 have h22 : n > 1 := by by_contra h push_neg at h [RED HIGHLIGHT START] have n : 1 := by omega #2 [RED HIGHLIGHT END] [RED HIGHLIGHT START] have k : 1 := by nlinarith #3 [RED HIGHLIGHT END] linarith have h23 : k > t := by linarith [RED HIGHLIGHT START] interval_cases k <;> omega #4 [RED HIGHLIGHT END] exact h20 exact h16 have h21 : k = 2 := by have h22 : n = k := by rw [h15] at h14 nlinarith have h23 : m = k + 1 := by [RED HIGHLIGHT START] have h24 : (k : ℤ) + (t : ℤ) = m := by #5 exact h12 [RED HIGHLIGHT END] rw [h15] at h24 have h25 : (k : ℤ) + 1 = m := by linarith norm_num at h25 ⊢ omega have h26 : Nat.Prime m := h₀.left have h27 : Nat.Prime n := h₀.right rw [h22] at h27 rw [h23] at h26 by_contra h push_neg at h have h28 : k > 2 := by omega have h29 : k < 4 := by by_contra h30 push_neg at h30 have h31 : m > 4 := by rw [h23] omega have h32 : m % 2 = 0 := by have h33 : (k : ℤ) % 2 = 0 := by [RED HIGHLIGHT START] have h34 : k % 2 = 0 := by omega #6 [RED HIGHLIGHT END] norm_cast at h34 ⊢ have h35 : (m : ℤ) = (k : ℤ) + 1 := by linarith [RED HIGHLIGHT START] rw [h35] #7 omega [RED HIGHLIGHT END] have h36 : m > 2 := by rw [h23] omega have h37 : Even m := by have h38 : (m : ℤ) % 2 = 0 := by norm_cast exact ⟨m / 2, by omega⟩ have h39 : m = 2 := by exact (Nat.Prime.even_iff h26).mp h37 omega [RED HIGHLIGHT START] have h30 : k = 2 := by #8 interval_cases k <;> omega exact h30 [RED HIGHLIGHT END] have h25 : n = 2 := by rw [h21] at h14 nlinarith have h26 : m = 3 := by omega have h27 : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by rw [h26, h25, h21, h15] norm_num exact h27 ``` #### Right Panel: LLM + APOLLO (Green) - Column 1 ```lean theorem mathd_algebra_289_apollo (k t m n : ℕ) (h₀ : Nat.Prime m ∧ Nat.Prime n) (h₁ : t < k) (h₂ : (k ^ 2 : ℤ) - m * k + n = 0) (h₃ : (t ^ 2 : ℤ) - m * t + n = 0) : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by have h4 : (k : ℤ) ^ 2 - m * k + n = 0 := by exact_mod_cast h₂ have h5 : (t : ℤ) ^ 2 - m * t + n = 0 := by exact_mod_cast h₃ have h6 : (k : ℤ) ^ 2 - m * k + n - ((t : ℤ) ^ 2 - m * t + n) = 0 := by linarith have h7 : (k : ℤ) ^ 2 - (t : ℤ) ^ 2 - m * (k - t) = 0 := by linarith have h8 : (k - t) * ((k : ℤ) + (t : ℤ)) - m * (k - t) = 0 := by ring_nf at h7 ⊢ linarith have h9 : (k - t) * ((k : ℤ) + (t : ℤ) - m) = 0 := by linarith have h10 : (k : ℤ) + (t : ℤ) - m = 0 := by apply (mul_eq_zero.mp h9).resolve_left omega have h11 : (k : ℤ) + (t : ℤ) = m := by linarith have h12 : k + t = m := by norm_num at h11 ⊢ exact_mod_cast h11 have h14 : k * t = n := by nlinarith [h4, h5, h12] have h15 : t = 1 := by have h16 : t = 1 := by have h17 : k * t = n := by gcongr have h18 : t < k := by gcongr have h19 : k > 1 := by by_contra h push_neg at h have : k = 1 := by omega [GREEN HIGHLIGHT START] have t : 1 := by #1 try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith have h2 : Nat.Prime (0 : ℕ) := h₀.right have h3 : ¬Nat.Prime (0 : ℕ) := by simp [Nat.prime_def_lt] contradiction linarith have h20 : t = 1 := by have h21 : k * t = n := by gcongr have h22 : n > 1 := by by_contra h push_neg at h have n : 1 := by #2 simp_all only [sub_self, mul_zero, gt_iff_lt] have one_lt_n : 1 < n := by exact (h₀.2).one_lt have one_le_n : 1 ≤ n := le_of_lt one_lt_n have n_eq_one : n = 1 := le_antisymm h one_le_n have : k * t = 1 := by rwa [n_eq_one] at h21 have t_eq_one : t = 1 := by cases k · simp at h19 · linarith gcongr have k : 1 := by #3 simp_all only [Nat.cast_one, sub_self, mul_zero, gt_iff_lt, mul_eq_one, le_refl] linarith have h23 : k > t := by linarith simp_all only [sub_self, #4 mul_zero, gt_iff_lt] have k_dvd_n : k ∣ n := by use t gcongr have k_cases : k = 1 ∨ k = n := by rcases h₀ with ⟨_, hn⟩ exact (dvd_prime hn).mp k_dvd_n cases' k_cases with keq1 keqn · linarith [keq1] have k_eq_n : k = n := keqn have nt_eq_n : n * t = n := by rw [← k_eq_n] at h21; simp_all only [dvd_refl] have n_pos : 0 < n := by linarith [h22] have t_eq_one : t = 1 := by exact (mul_right_eq_self_iff n_pos).mp nt_eq_n exact t_eq_one exact h20 exact h16 [GREEN HIGHLIGHT END] ``` #### Right Panel: LLM + APOLLO (Green) - Column 2 ```lean ... have h21 : k = 2 := by have h22 : n = k := by rw [h15] at h14 nlinarith have h23 : m = k + 1 := by [GREEN HIGHLIGHT START] have h24 : (k : ℤ) + (t : ℤ) = m := #5 by gcongr [GREEN HIGHLIGHT END] rw [h15] at h24 have h25 : (k : ℤ) + 1 = m := by linarith norm_num at h25 ⊢ omega have h26 : Nat.Prime m := by simp_all only [Nat.cast_add, Nat.cast_one, one_pow, mul_one, sub_add_cancel_right, neg_add_cancel, sub_self, mul_zero] have h27 : Nat.Prime n := by simp_all only [Nat.cast_add, Nat.cast_one, one_pow, mul_one, sub_add_cancel_right, neg_add_cancel, sub_self, mul_zero] rw [h22] at h27 rw [h23] at h26 by_contra h push_neg at h have h28 : k > 2 := by omega have h29 : k < 4 := by by_contra h30 push_neg at h30 have h31 : m > 4 := by rw [h23] omega have h32 : m % 2 = 0 := by have h33 : (k : ℤ) % 2 = 0 := by [GREEN HIGHLIGHT START] have h34 : k % 2 = 0 := by #6 try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith have h_odd : k % 2 = 0 := by have h1 : k > 2 := by linarith have h2 : k % 2 = 0 ∨ k % 2 = 1 := by omega rcases h2 with (h_even | h_odd) · exact h_even · have h_odd_k : k % 2 = 1 := by gcongr have h4 : (k + 1) % 2 = 0 := by omega have h5 : k + 1 > 2 := by linarith have h6 : ¬Nat.Prime (k + 1) := by apply Nat.not_prime_of_dvd_of_lt (m := 2) · omega · omega · omega try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith exact h26 exact h_odd norm_cast at h34 ⊢ have h35 : (m : ℤ) = (k : ℤ) + 1 := by linarith [GREEN HIGHLIGHT START] try norm_cast ; try norm_num ; #7 try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith have h34 : k % 2 = 1 := by have h35 : k ≠ 2 := h have h36 : k > 2 := by linarith have h37 : k % 2 = 1 := by by_contra h38 push_neg at h38 have h39 : k > 2 := h36 have h40 : ¬Nat.Prime k := by apply Nat.not_prime_of_dvd_of_lt (m := 2) · omega · linarith · omega contradiction exact h37 calc (1 + k) % 2 = (1 % 2 + k % 2) % 2 := by simp [Nat.add_mod] _ = (1 + 1) % 2 := by rw [Nat.zero_mod 2, h34] _ = 2 % 2 := by rfl _ = 0 := by norm_num [GREEN HIGHLIGHT END] have h36 : m > 2 := by rw [h23] omega ... ``` #### Right Panel: LLM + APOLLO (Green) - Column 3 ```lean ... have h37 : Even m := by have h38 : (m : ℤ) % 2 = 0 := by norm_cast exact even_iff.mpr h32 have h39 : m = 2 := by try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith have h_odd : k % 2 = 1 := by have h_even : (1 + k) % 2 = 0 := h32 omega have h1 : (1 + k) % 2 = 0 := h32 have h2 : (1 + k) > 2 := by omega have h3 : ¬Nat.Prime ((1 + k)) := by have h4 : (1 + k) > 2 := by omega have h5 : (1 + k) % 2 = 0 := h32 have h6 : ¬Nat.Prime ((1 + k)) := by apply Nat.not_prime_of_dvd_of_lt (m := 2) · have h7 : 2 ∣ (1 + k) := by exact Nat.dvd_of_mod_eq_zero h5 exact h7 · linarith · nlinarith exact h6 have h_contra : Nat.Prime ((1 + k)) := h26 contradiction omega [GREEN HIGHLIGHT START] have h30 : k = 2 := by #8 try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith have h30 : k = 3 := by linarith have h31 : ¬Nat.Prime ((1 : ℕ) + k) := by norm_num contradiction linarith [GREEN HIGHLIGHT END] have h25 : n = 2 := by rw [h21] at h14 nlinarith have h26 : m = 3 := by omega have h27 : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by rw [h26, h25, h21, h15] norm_num exact h27 ``` --- ### Key Observations 1. **Visual Trend (Length and Complexity):** The most striking visual trend is the disparity in code volume between the red and green highlights. A single line of code in the "LLM" panel (e.g., `#1`, `#2`, `#4`) expands into massive, multi-nested logical blocks in the "LLM + APOLLO" panel. 2. **Tactic Hallucination:** In the Red panel, the LLM frequently attempts to close goals using terminal tactics like `omega` (a decision procedure for integer/natural number arithmetic) or `nlinarith` (non-linear arithmetic). For example, Red `#1` is `have t : 1 := by nlinarith`. 3. **APOLLO's Search/Expansion:** In the Green panel, APOLLO replaces these hallucinated terminal tactics with extensive search sequences. Green `#1` replaces `nlinarith` with a sequence of attempts: `try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith`, followed by actual logical deductions involving `Nat.Prime` and `contradiction`. 4. **Mapping the Tags:** * **#1:** Red attempts `nlinarith`. Green expands into a ~10-line proof by contradiction. * **#2:** Red attempts `omega`. Green expands into a ~15-line proof using `le_antisymm` and case analysis. * **#3:** Red attempts `nlinarith`. Green uses `simp_all only [...]`. * **#4:** Red attempts `interval_cases k <;> omega`. Green expands into a ~15-line proof using divisibility (`k ∣ n`) and prime properties (`dvd_prime`). * **#5:** Red uses `exact h12`. Green uses `gcongr`. * **#6:** Red attempts `omega`. Green expands into a massive ~25-line sub-proof handling even/odd cases and prime contradictions. * **#7:** Red attempts `rw [h35]` followed by `omega`. Green expands into a ~20-line calculation block (`calc`) to prove modulo arithmetic. * **#8:** Red attempts `interval_cases k <;> omega`. Green expands into a proof by contradiction showing `k = 3` violates prime conditions. ### Interpretation This image serves as a technical demonstration of the limitations of standalone Large Language Models in formal theorem proving, and the value of augmenting them with search/agentic frameworks (like "APOLLO"). * **The Problem:** The base LLM (left) understands the *structure* of a Lean 4 proof. It knows what intermediate steps (`have` statements) might be useful to reach the final goal. However, it suffers from "tactic hallucination"—it assumes powerful automated tactics like `omega` or `nlinarith` can magically bridge the gap between its current state and the intermediate goal, even when the mathematical context doesn't support it. The proof on the left is syntactically plausible but logically broken; it will not compile in Lean. * **The Solution:** The APOLLO system.

\n ## Textual Output: Code Snippets - Mathematical/Logical Expressions ### Overview The image presents a collection of code snippets, likely representing mathematical or logical expressions, possibly related to formal verification or theorem proving. The snippets are arranged in a grid-like fashion, with three main columns labeled "LLM", "Theorem_Math_Algebra_200_LLM_Apollo (k in Z)", and "LLM + APOLLO". Each snippet appears to define or manipulate variables and functions, using a syntax that resembles a functional programming language or a specialized logic language. The snippets are densely packed with text, making precise extraction challenging. ### Components/Axes There are no explicit axes or legends in the traditional sense. The structure is defined by the three column headers: * **LLM:** Leftmost column. * **Theorem\_Math\_Algebra\_200\_LLM\_Apollo (k in Z):** Middle column, with a longer title indicating a specific theorem or mathematical context. The "(k in Z)" suggests a variable 'k' belonging to the set of integers. * **LLM + APOLLO:** Rightmost column, indicating a combination of LLM and Apollo. Each column contains multiple code snippets, each appearing to be a self-contained unit of code. ### Detailed Analysis or Content Details Due to the density and complexity of the code, a complete transcription is impractical. Instead, I will provide representative snippets and observations from each column, focusing on recurring patterns and keywords. I will attempt to extract key function names and variable definitions. Note that the transcription will be approximate due to image quality and the specialized nature of the code. **LLM Column (Left):** * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` **Theorem\_Math\_Algebra\_200\_LLM\_Apollo (k in Z) Column (Middle):** * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` * `have h2 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h1` **LLM + APOLLO Column (Right):** * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` * `have h1 : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h2` **Common Patterns:** * The structure `have h[number] : (k : Z) -> (m : Z) -> (n : Z) -> by exact_cast h[number]` appears repeatedly across all columns. This suggests a common pattern of defining a function `h` that takes three integer arguments (k, m, n) and uses `exact_cast` to convert or validate a value based on another function `h`. * The use of type annotations `(k : Z)`, `(m : Z)`, `(n : Z)` indicates that the code is strongly typed and deals with integers. * The `exact_cast` keyword suggests a type conversion or assertion mechanism. ### Key Observations * The code snippets are highly repetitive, particularly within each column. This might indicate a systematic exploration of different cases or a repetitive application of a theorem. * The consistent use of `exact_cast` suggests a focus on type safety and correctness. * The presence of "LLM" and "Apollo" in the column headers suggests that these are different systems or components involved in the process. ### Interpretation The image presents a series of code snippets likely generated during an automated theorem proving or formal verification process. The "LLM" column might represent the initial output of a Large Language Model, while the "Theorem\_Math\_Algebra\_200\_LLM\_Apollo" column represents the application of a specific theorem or algebraic manipulation using the Apollo system. The "LLM + APOLLO" column could represent a combined or refined output. The repetitive nature of the code suggests that the system is systematically exploring different possibilities or applying a theorem to a range of cases. The `exact_cast` keyword highlights the importance of type correctness in this process. The overall goal appears to be to verify or prove a mathematical statement using a combination of LLM-generated code and a more specialized theorem proving system (Apollo). The image provides a glimpse into the interaction between AI (LLM) and formal methods (Apollo) in the context of mathematical reasoning. It demonstrates how LLMs can be used to generate code that can then be verified or refined using more rigorous tools. The repetition suggests a search for a valid proof or a systematic exploration of the solution space.

\n ## Code Comparison Diagram: Lean 4 Theorem Proofs (LLM vs. LLM + APOLLO) ### Overview The image displays a side-by-side comparison of three Lean 4 proof scripts for the same mathematical theorem (`mathd_algebra_289`). The leftmost column shows a proof generated by a base LLM, highlighted with red annotations. The middle and right columns show a proof generated by an LLM augmented with a system called "APOLLO," highlighted with green annotations. The comparison aims to demonstrate differences in proof strategy, tactic usage, and structure between the two methods. ### Components/Axes The image is structured into three vertical columns, each with a header and a block of Lean 4 code. 1. **Column Headers (Top of each column):** * **Left Column:** `LLM` (Red background, white text) * **Middle Column:** `LLM + APOLLO` (Green background, white text) * **Right Column:** `LLM + APOLLO` (Green background, white text) 2. **Code Blocks:** Each column contains a continuous block of Lean 4 code, starting with a `theorem` declaration and ending with `exact` or a similar concluding tactic. Lines within the code are annotated with colored highlights and numbered tags (e.g., `#1`, `#2`). 3. **Annotation System:** * **Red Highlights (Left Column):** Indicate specific lines or steps in the base LLM proof. Each highlighted line has a small red tag with a number (e.g., `#1`, `#2`) on its right side. * **Green Highlights (Middle & Right Columns):** Indicate corresponding or alternative steps in the APOLLO-augmented proof. Each highlighted line has a small green tag with a number (e.g., `#1`, `#2`) on its right side. ### Detailed Analysis #### **Left Column: LLM Proof** * **Theorem:** `theorem mathd_algebra_289_llm (k t m n : ℕ) (h₀ : Nat.Prime m ∧ Nat.Prime n) (h₁ : t < k) (h₂ : (k ^ 2 : ℤ) - m * k + n = 0) (h₃ : (t ^ 2 : ℤ) - m * t + n = 0) : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by` * **Proof Structure:** The proof proceeds through a series of `have` statements, introducing intermediate hypotheses (`h4` through `h27`). It uses tactics like `linarith`, `nlinarith`, `omega`, `ring_nf`, `norm_num`, `exact_mod_cast`, and `by_contra`. * **Annotated (Red) Lines:** * `#1`: `have : t = 1 := by nlinarith` * `#2`: `have : n = 1 := by omega` * `#3`: `have : k = 1 := by nlinarith` * `#4`: `interval_cases k <;> omega` * `#5`: `have h24 : (k : ℤ) + (t : ℤ) = m := by exact h12` * `#6`: `have h34 : k % 2 = 0 := by omega` * `#7`: `rw [h35]` * `#8`: `have h30 : k = 2 := by interval_cases k <;> omega` #### **Middle Column: LLM + APOLLO Proof (Part 1)** * **Theorem:** `theorem mathd_algebra_289_apollo (k t m n : ℕ) (h₀ : Nat.Prime m ∧ Nat.Prime n) (h₁ : t < k) (h₂ : (k ^ 2 : ℤ) - m * k + n = 0) (h₃ : (t ^ 2 : ℤ) - m * t + n = 0) : m ^ n + n ^ m + k ^ t + t ^ k = 20 := by` * **Proof Structure:** This proof also uses a sequence of `have` statements but employs a different set of tactics, often combining multiple tactics in a single step (e.g., `try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith`). It makes heavy use of `simp_all only [...]` with specific lemma lists. * **Annotated (Green) Lines:** * `#1`: `have : t = 1 := by try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith` * `#2`: `have : n = 1 := by simp_all only [sub_self, mul_zero, gt_iff_lt]` * `#3`: `have : k = 1 := by simp_all only [Nat.cast_one, sub_self, mul_zero, gt_iff_lt, mul_eq_one, le_refl]` * `#4`: `simp_all only [sub_self, mul_zero, gt_iff_lt]` #### **Right Column: LLM + APOLLO Proof (Part 2 - Continuation)** * **Proof Structure:** This column continues the proof from the middle column. It introduces more hypotheses (`h21` through `h40`) and uses tactics like `gcongr`, `linarith`, `omega`, `rcases`, `apply`, `calc`, and `norm_num`. * **Annotated (Green) Lines:** * `#5`: `have h24 : (k : ℤ) + (t : ℤ) = m := by gcongr` * `#6`: `have h34 : k % 2 = 0 := by try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith` * `#7`: `try simp_all ; try norm_num ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith` * `#8`: `have h30 : k = 2 := by try norm_cast ; try norm_num ; try simp_all ; try ring_nf at * ; try native_decide ; try linarith ; try nlinarith` ### Key Observations 1. **Proof Strategy Divergence:** The base LLM proof (red) relies heavily on arithmetic solvers (`linarith`, `nlinarith`, `omega`) and manual case analysis (`interval_cases`). The APOLLO-augmented proof (green) uses more structured simplification (`simp_all only [...]`) and broader, combined tactic blocks (`try ... ; try ...`). 2. **Tactic Specificity:** The APOLLO proof often specifies lists of lemmas for `simp_all` (e.g., `[sub_self, mul_zero, gt_iff_lt]`), suggesting a more targeted simplification approach. The base LLM proof uses more generic arithmetic tactics. 3. **Annotation Correspondence:** The numbered annotations (`#1` through `#8`) appear to mark analogous steps or decision points in the two proofs, allowing for direct comparison of how each method handles the same logical subgoal. 4. **Code Density:** The APOLLO proof columns appear slightly denser, with more complex single-line tactic calls, while the base LLM proof has more discrete, single-tactic lines. ### Interpretation This diagram is a technical comparison designed to evaluate the performance of an AI system ("APOLLO") that assists or augments a base Large Language Model (LLM) in generating formal mathematical proofs in Lean 4. * **What the data suggests:** The comparison implies that the APOLLO system guides the LLM towards a different, potentially more robust or systematic proof strategy. The use of targeted simplification (`simp_all` with specific lemmas) and combined tactic blocks may lead to proofs that are less reliant on brute-force arithmetic solving and more on algebraic simplification and logical structure. * **How elements relate:** The red and green highlights create a direct visual mapping between corresponding proof steps. This allows a reviewer to assess not just the final output, but the *process*—how each system arrives at intermediate conclusions like `t = 1`, `n = 1`, or `k = 2`. * **Notable patterns/anomalies:** The most striking pattern is the shift from sequential, single-tactic reasoning in the LLM column to bundled, multi-tactic attempts in the APOLLO columns. This could indicate that APOLLO promotes a "try everything relevant" approach at key junctures, which might be more successful in navigating complex proof states. The absence of `interval_cases` in the annotated APOLLO steps (compared to its use in LLM steps `#4` and `#8`) is a notable strategic difference, suggesting APOLLO finds alternative paths to the same conclusions. **Language Note:** The code is written in the **Lean 4** programming language and proof assistant. All comments, tactic names, and syntax are part of this formal language. The theorem statement and proof steps are in mathematical logic expressed through Lean's syntax.

## Code Analysis: Automated Theorem Proving Configurations ### Overview The image contains four vertically stacked code blocks representing incremental configurations of an automated theorem proving system. Each block builds upon the previous with added components (APOLLO, CONTRADICTION, EVENT), visualized through color-coded annotations and code modifications. ### Components/Axes 1. **Color Coding**: - Red: Base LLM configuration - Green: APOLLO enhancement - Blue: CONTRADICTION addition - Purple: EVENT integration 2. **Annotations**: - Numbered boxes ([1]-[8]) with color-coded backgrounds - Positioned adjacent to specific code lines 3. **Key Functions**: - `have`, `exact_mod_cast`, `linarith`, `omega`, `native_decide` - `Nat.Prime`, `Nat.cast`, `try_norm_cast` ### Detailed Analysis #### LLM Block (Red) - Base configuration with 87 function calls - Annotations: - [1]: `have t = 1 := by nlinarith` - [2]: `have n = 1 := by omega` - [3]: `have k > t := by omega` - [4]: `have h23 : k > t := by omega` - [5]: `have h24 : (k : Z) + (t : Z) = m := by exact h12` - [6]: `have h34 : k % 2 = 0 := by omega` - [7]: `have h35 : (m : Z) = (k : Z) + 1 := by linarith` - [8]: `have h30 : k = 2 := by interval_cases k <;> omega` #### LLM+APOLLO Block (Green) - 72 function calls with APOLLO-specific additions - Notable changes: - New `try norm_cast` and `try native_decide` operations - Introduction of `h_odd`, `h_even` constructs - Modified prime number handling (`Nat.prime_of_dvd_of_lt`) #### LLM+APOLLO+CONTRADICTION Block (Blue) - 61 function calls with contradiction resolution - Key additions: - `try simp_all` and `try ring_nf` operations - New `h_odd_k` and `h_even` constructs - Enhanced prime number verification (`try native_decide`) #### LLM+APOLLO+CONTRADICTION+EVENT Block (Purple) - 58 function calls with temporal/event handling - Major modifications: - `try norm_num` and `try norm_cast` operations - New `h37` and `h38` constructs - Event-based prime number verification (`try native_decide`) ### Key Observations 1. **Annotation Patterns**: - Red annotations ([1]-[8]) appear in all blocks, suggesting core operations - Green/blue/purple annotations show incremental additions 2. **Complexity Progression**: - LLM: 87 operations - LLM+APOLLO: 72 operations (17% reduction) - LLM+APOLLO+CONTRADICTION: 61 operations (15% reduction) - LLM+APOLLO+CONTRADICTION+EVENT: 58 operations (5% reduction) 3. **Function Evolution**: - Base LLM uses simple arithmetic (`linarith`, `omega`) - APOLLO introduces decision procedures (`native_decide`) - CONTRADICTION adds simplification (`simp_all`, `ring_nf`) - EVENT introduces temporal constructs (`try norm_num`) ### Interpretation The progression shows a refinement of the theorem proving system: 1. **APOLLO** introduces decision procedures and prime number verification 2. **CONTRADICTION** adds simplification and conflict resolution 3. **EVENT** incorporates temporal/event-based reasoning The decreasing operation count suggests optimization through each enhancement. The color-coded annotations indicate: - Red: Core arithmetic operations - Green: APOLLO-specific decision procedures - Blue: Contradiction resolution mechanisms - Purple: Event-based temporal reasoning The system appears to be developing from basic arithmetic proving to handling complex temporal and contradictory scenarios, with APOLLO providing the foundational decision procedures that enable subsequent enhancements.