## Flowchart: Apollo Algorithm
### Overview
The image depicts a technical flowchart titled "Apollo Algorithm," illustrating a multi-step process for proof validation and assembly. The workflow includes components like a Syntax Refiner, Sorrifier, Auto Solver, Proof Assembler, and LLM (Large Language Model). Decision points and code snippets (labeled "Invalid Proof" and "Correct Proof") are integrated into the diagram.
### Components/Axes
1. **Main Workflow Components**:
- **Syntax Refiner** (icon: gear/settings)
- **Sorrifier** (icon: hammer)
- **Auto Solver** (icon: pencil)
- **Proof Assembler** (icon: trowel/brick wall)
- **LLM** (network diagram with nodes)
2. **Decision Points**:
- "number of attempts > r?" (diamond shape, red)
- "Does the proof compile in Lean?" (diamond shape, green)
3. **Code Snippets**:
- **Invalid Proof** (red box, contains errors like `h: a % 10 = 0`)
- **Correct Proof** (green box, fixes errors with adjusted conditions)
### Detailed Analysis
- **Syntax Refiner** → **Sorrifier** → **Auto Solver**: Sequential flow with feedback loops.
- **Proof Assembler** uses a brick-wall metaphor, suggesting incremental construction.
- **LLM** receives extracted subproofs and assembles the final proof.
- **Decision Points**:
- If "number of attempts > r?" is **YES**, the process loops back to the Syntax Refiner.
- If "Does the proof compile in Lean?" is **YES**, the proof proceeds to the LLM.
- **Code Snippets**:
- **Invalid Proof** includes errors like `h: a % 10 = 0` and `h: b % 10 = 4`.
- **Correct Proof** replaces these with `h: a % 10 = 6` and `h: b % 10 = 6`, resolving contradictions.
### Key Observations
1. **Feedback Loops**: The process iterates until the proof compiles in Lean or exceeds the attempt limit.
2. **Error Correction**: The green "Correct Proof" explicitly addresses contradictions present in the red "Invalid Proof."
3. **Visual Metaphors**: The brick-wall analogy for the Proof Assembler emphasizes modular, step-by-step construction.
### Interpretation
The Apollo Algorithm appears to automate formal proof validation and repair. The workflow prioritizes iterative refinement (via the Sorrifier and Auto Solver) and leverages an LLM for final assembly. The code snippets highlight common pitfalls (e.g., incorrect modulo operations) and their resolutions, suggesting the algorithm’s focus on arithmetic and logical consistency. The decision points imply a balance between computational effort (attempt limits) and correctness (Lean compilation).
**Notable Patterns**:
- The use of color-coded code blocks (red/green) visually distinguishes error states from validated states.
- The brick-wall metaphor for the Proof Assembler reinforces the idea of building proofs incrementally, akin to masonry.
- The LLM’s role as a final assembler suggests integration of machine learning for complex proof synthesis.
