## Code Snippet Analysis: LLM, LLM+APOLLO, and APOLLO Theorem Proofs ### Overview The image contains three side-by-side code blocks demonstrating formal theorem proofs in a proof assistant environment (likely Coq or Lean). Each block represents a different configuration: pure LLM, LLM with APOLLO integration, and pure APOLLO. The code focuses on real number arithmetic properties with specific attention to logarithmic operations and linear arithmetic. ### Components/Axes 1. **Imports**: - All blocks import `Mathlib` and `Aesop` libraries - APOLLO version includes additional imports for linear arithmetic 2. **Configuration**: - `set_option maxHeartbeats 0` in all blocks - `open BigOperators Real Nat Topology Rat` in all blocks 3. **Theorem Structure**: - LLM: `theorem amc12a_2019_p12_llm` - LLM+APOLLO: `theorem amc12a_2019_p12_apollo` - APOLLO: `theorem amc12a_2019_p12_apollo` (different proof approach) 4. **Highlighted Sections**: - Red highlights in LLM block (lines 10-13) - Green highlights in LLM+APOLLO (lines 10-13, 22-25) - Green highlight in APOLLO (lines 10-13) ### Detailed Analysis **LLM Block**: - Proves: `x ≠ 0 → y ≠ 0 → (log(x/y)/log(2))^2 = 20` - Key steps: - `have x_nonzero : x ≠ 0` (line 10) - `have y_nonzero : y ≠ 0` (line 14) - Uses `rfl` (reflexivity) and `simp` (simplification) tactics **LLM+APOLLO Block**: - Same theorem but with APOLLO integration - Additional steps: - `linarith` (line 18) for linear arithmetic - `field_simp` with extended field operations - Uses `rw` (rewrite) with complex expressions **APOLLO Block**: - Pure APOLLO implementation - Key differences: - Uses `linarith` instead of `rfl` - More complex field simplification - Additional `rw` steps with logarithmic properties ### Key Observations 1. All implementations share core structure but differ in proof tactics 2. APOLLO versions show more sophisticated handling of: - Logarithmic properties (`log_mul`, `log_inv`) - Linear arithmetic (`linarith`) - Field operations (`mul_comm`) 3. Highlighted sections focus on: - Non-zero assumptions (`x_nonzero`, `y_nonzero`) - Logarithmic identity proofs - Field simplification steps ### Interpretation The code demonstrates formal verification of logarithmic identities in real numbers. The progression from LLM to APOLLO shows: 1. **Tactical Evolution**: APOLLO enables more powerful proof strategies 2. **Complexity Tradeoff**: APOLLO proofs are more verbose but potentially more general 3. **Logarithmic Focus**: All implementations center on `log(x/y)` properties 4. **Formal Rigor**: Explicit handling of division by zero and field properties The APOLLO integration appears to provide: - Enhanced linear arithmetic capabilities - More sophisticated field operation handling - Better support for complex rewrite rules This suggests APOLLO adds significant value for formal verification of mathematical properties involving real numbers and logarithms, particularly in handling complex arithmetic expressions and maintaining proof correctness through automated tactics.