## 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.