## Screenshot: Coq Theorem Proof Comparison (LLM vs LLM + APOLLO) ### Overview The image shows two side-by-side Coq theorem proofs comparing the original LLM implementation (left) with an APOLLO-enhanced version (right). Both prove the same algebraic property but differ in implementation details and proof strategies. ### Components/Axes - **Left Panel (LLM)**: - Theorem: `mathd_algebra_158_llm` - Hypotheses: `h0 : Even a`, `h1 : ∀k ∈ Finset.range 8, 2*k+1 ∈ Finset.range 5` - Key steps: Manual case analysis with `linarith`, explicit division checks - Highlighted sections: - `#1`: `have h_even : ∃m, a = 2*m := h0` - `#2`: `have two_divides_53 : ∃d, 53 = 2*d := by ...` - `#3`: `rw [mul_mod] at mod53` - **Right Panel (LLM + APOLLO)**: - Theorem: `mathd_algebra_158_apollo` - Hypotheses: `h0 : Even a` - Key steps: - Simplified case analysis with `by omega` - Automated division checks - Commented-out `rw [mul_mod] at mod53` (line 15) - Highlighted sections: - `#1`: `have h_even : ∃m, a = 2*m := by omega` - `#2`: `have two_divides_53 : ∃d, 53 = 2*d := by omega` ### Content Details **LLM Version**: 1. Explicit case analysis for `k` values 2. Manual verification of `53 = 2*d` using `linarith` 3. Explicit modular reduction with `rw [mul_mod]` **APOLLO Version**: 1. Uses `by omega` for automatic case splitting 2. Leverages APOLLO's division reasoning (`by omega`) 3. Commented-out manual modular reduction step ### Key Observations 1. APOLLO version reduces proof length by 40% through automation 2. Manual case analysis (`linarith`) replaced with `omega` tactic 3. APOLLO introduces automated division reasoning (`exact two_divides_53`) 4. Commented-out `rw [mul_mod]` suggests potential optimization path ### Interpretation The APOLLO enhancement demonstrates how automated reasoning can: 1. Reduce proof maintenance effort through case generalization 2. Improve proof portability across different ring structures 3. Enable more concise expression of arithmetic properties 4. Potentially identify optimization opportunities (as shown by the commented-out `rw`) The comparison highlights APOLLO's ability to: - Automate case splitting in linear arithmetic - Handle division properties without explicit case analysis - Maintain proof correctness while reducing human intervention - Provide clear paths for further optimization through commented code