\n ## Diagram: Side-by-Side Code Comparison (LLM vs. LLM + APOLLO) ### Overview The image displays a side-by-side comparison of two blocks of formal mathematical proof code written in the Lean 4 language. The left panel, titled "LLM" in a red header, shows a proof generated by a Large Language Model. The right panel, titled "LLM + APOLLO" in a green header, shows a more detailed and structured proof, presumably generated or enhanced by a system called APOLLO. The comparison highlights differences in proof strategy, detail, and the use of automated tactics. ### Components/Axes The image is divided into two vertical panels of equal width. **Left Panel (LLM):** * **Header:** A red rectangular box at the top containing the text "LLM". * **Code Block:** A block of Lean 4 code on a light pink background. * **Annotations:** Two highlighted sections in pink, labeled `#1` and `#2`. **Right Panel (LLM + APOLLO):** * **Header:** A green rectangular box at the top containing the text "LLM + APOLLO". * **Code Block:** A block of Lean 4 code on a light green background. * **Annotations:** Two highlighted sections in green, labeled `#1` and `#2`. The `#1` annotation is significantly more extensive than its counterpart on the left. ### Detailed Analysis **1. Common Code Structure (Both Panels):** Both proofs begin with identical setup lines: ```lean import Mathlib import Aesop set_option maxHeartbeats 0 open BigOperators Real Nat Topology Rat theorem induction_pord1p1on2powklt5on2_llm (n k : ℕ) (h₀ : 0 < n) : (∏ k in Finset.Icc 1 n, 1 + (1 : ℝ) / 2 ^ k) < 5 / 2 := by ``` This defines a theorem named `induction_pord1p1on2powklt5on2_llm` (the right panel uses `_apollo`). The theorem states that for natural numbers `n` and `k`, with the hypothesis `h₀ : 0 < n`, the product of the terms `(1 + 1/2^k)` for `k` from 1 to `n` is strictly less than 5/2. **2. Proof Strategy - Left Panel (LLM):** The proof proceeds by induction on `n`. ```lean induction n with | zero => contradiction | succ n hn => cases n with | zero => norm_num #1 -- Annotation #1 | succ n => simp_all [Finset.prod_Icc_succ_top, Nat.cast_succ, div_eq_mul_inv] norm_num #2 -- Annotation #2 ring_nf linarith ``` * **Base Case (`zero`):** Discharged by `contradiction` (since `h₀ : 0 < 0` is false). * **Inductive Step (`succ n hn`):** Uses `cases n` to split into two subcases. * Subcase `zero` (i.e., `n=1`): Uses `norm_num` (Annotation #1) to verify the numeric inequality. * Subcase `succ n` (i.e., `n ≥ 2`): Uses `simp_all` with specific lemmas, then `norm_num` (Annotation #2), `ring_nf` for normalization, and `linarith` for linear arithmetic. **3. Proof Strategy - Right Panel (LLM + APOLLO):** The proof also begins with induction on `n` but introduces a nested induction on `k` within the inductive step. ```lean induction n with | zero => contradiction | succ n hn => cases n with | zero => norm_num induction k with -- Nested induction begins | zero => norm_num [Finset.prod_Icc_succ_top] at hn h₀ <;> linarith | succ k ih => cases k with | zero => norm_num [Finset.prod_Icc_succ_top] at hn h₀ <;> linarith | succ k => simp_all [Finset.prod_Icc_succ_top, pow_succ, mul_comm] linarith -- End of nested induction block (Annotation #1) | succ n => simp_all [Finset.prod_Icc_succ_top, Nat.cast_succ, div_eq_mul_inv] norm_num #2 -- Annotation #2 ring_nf linarith ``` * **Base Case (`zero`):** Identical to the left panel. * **Inductive Step (`succ n hn`):** Uses `cases n`. * Subcase `zero` (i.e., `n=1`): This is where the major difference lies (Annotation #1). Instead of a single `norm_num`, it performs a **nested induction on `k`**. This inner induction has its own base case (`zero`) and inductive step (`succ k ih`), which is further split. This suggests a more granular, step-by-step verification of the product inequality for the specific case where `n=1`, possibly to handle the product over `k` more explicitly. * Subcase `succ n` (i.e., `n ≥ 2`): The code is identical to the left panel's corresponding subcase, including Annotation #2. **4. Annotations:** * **Annotation #1 (Left - LLM):** A single tactic: `norm_num`. * **Annotation #1 (Right - LLM + APOLLO):** A multi-line block implementing a nested induction on `k` with several tactics (`norm_num`, `induction`, `cases`, `simp_all`, `linarith`). This is the core differentiator. * **Annotation #2 (Both Panels):** Identical in both: `norm_num`. ### Key Observations 1. **Structural Divergence:** The primary difference is in the subcase where `n=1`. The LLM proof uses a single tactic, while the LLM+APOLLO proof expands this into a detailed nested induction on the product index `k`. 2. **Proof Granularity:** The APOLLO-enhanced proof is more verbose and structured, breaking down the problem further. This may indicate a strategy to increase proof reliability or provide more detailed steps for verification. 3. **Identical Core Logic:** For the base case (`n=0`) and the inductive subcase for `n ≥ 2`, the proofs are identical. The enhancement is localized to a specific, potentially tricky part of the proof. 4. **Use of Tactics:** Both proofs use a similar set of Lean tactics (`induction`, `cases`, `norm_num`, `simp_all`, `linarith`), but APOLLO's version uses them in a more nested and explicit sequence. ### Interpretation This diagram illustrates a comparison between a standard LLM-generated formal proof and one augmented by a system named APOLLO. The key takeaway is that **APOLLO appears to enhance the proof by introducing more detailed, structured reasoning in complex subgoals.** * **What the data suggests:** The LLM alone produced a correct but minimally detailed proof for the `n=1` case. The APOLLO system identified this as a point requiring deeper justification and automatically generated a nested induction on `k` to provide that justification. This suggests APOLLO may function as a **proof refinement or debugging tool**, expanding high-level tactic calls into more fundamental steps. * **How elements relate:** The two panels are direct counterparts. The identical headers and theorem names establish they are solving the same problem. The divergence in the `| zero =>` subcase under `| succ n hn =>` is the focal point, demonstrating APOLLO's intervention. * **Notable patterns/anomalies:** The most significant pattern is the transformation of a single `norm_num` tactic into a multi-step inductive argument. This is not an error but a deliberate enrichment of the proof. It implies that for full formal verification or educational purposes, the APOLLO-style proof is more robust and transparent, as it leaves fewer steps to automated "black box" tactics. The fact that the rest of the proof remains unchanged suggests APOLLO's modifications are targeted and context-aware.