## Code Comparison: LLM vs LLM + APOLLO
### Overview
The image shows two code snippets implementing a mathematical theorem (`algebra_2varlineareq...`) using symbolic computation. The left side represents the base LLM implementation, while the right side shows the enhanced LLM + APOLLO version. Both use the `Mathlib` and `Aesop` libraries for formal verification.
### Components/Axes
- **Imports**:
- `Mathlib` (mathematical library)
- `Aesop` (automated theorem prover)
- **Options**:
- `maxHeartbeats 0` (execution timeout)
- `pp.numericTypes true` (enable numeric types)
- `pp.coercions.types true` (enable type coercions)
- **Context**:
- `open BigOperators Real Nat Topology Rat` (mathematical domain)
### Detailed Analysis
#### LLM Code (Left)
1. **Theorem Definition**:
- `algebra_2varlineareq...` defines a theorem with variables `f`, `z`, `h0`, `h1`, `h2`.
- Equations:
- `h0 := f + (3 : ℂ) * z = (11 : ℂ)`
- `h1 := (3 : ℂ) * (f - 1) - (5 : ℂ) * z = (-68 : ℂ)`
- Goal: Solve for `z` using `linarith` (linear arithmetic solver).
2. **Key Steps**:
- Simplify equations using `mul_sub` (multiplication/subtraction).
- Combine terms with `simpa` (simplification).
- Solve for `z` via `linarith` (highlighted in red).
- Substitute `z` back to solve for `f` (highlighted in red).
3. **Output**:
- Final equations: `f_eq : f = -10` and `z_eq : z = 7`.
#### LLM + APOLLO Code (Right)
1. **Enhanced Steps**:
- Adds normalization (`try_norm_cast`, `try_norm_num`) and ring normalization (`try_ring_nf`).
- Uses `native_decide` for decision-making in ring structures.
- Introduces `h_nonzero`, `h_div`, and `h_simpl` for handling non-zero elements and division.
2. **Key Differences**:
- More granular steps for normalization and type handling.
- Explicit checks for `h_nonzero` and `h_div` (highlighted in green).
- Uses `simp_all` and `ring_nf` for comprehensive simplification.
3. **Output**:
- Same final equations (`f_eq : f = -10`, `z_eq : z = 7`), but with additional intermediate steps for robustness.
### Key Observations
- **LLM**: Directly solves equations with minimal steps, relying on `linarith`.
- **LLM + APOLLO**: Adds normalization and decision-making steps, likely improving reliability in complex algebraic contexts.
- **Highlighted Lines**:
- Red: Critical equation-solving steps (`linarith`, substitution).
- Green: APOLLO-specific normalization and decision steps.
### Interpretation
The LLM + APOLLO version enhances the base LLM by incorporating formal verification techniques (e.g., normalization, ring arithmetic) to handle edge cases and ensure correctness. While both achieve the same result, APOLLO’s approach is more rigorous, reducing the risk of errors in symbolic computation. This aligns with APOLLO’s goal of bridging the gap between automated theorem proving and practical mathematical problem-solving.
