## Diagram: Proof Search and Tactic Generation Workflow
### Overview
The diagram illustrates a workflow for automated theorem proving, combining formal logic (proof search) and machine learning (tactic generation). It shows how a formal math library and language model collaborate to generate proof strategies.
### Components/Axes
- **Left Section (Proof Search)**:
- **Boxes**:
- `n : N` (natural number)
- `ih: add 0 n = n` (inductive hypothesis)
- `add 0 0 = 0` (base case)
- `add 0 (n+1) = n+1` (inductive step)
- `false` (failure case)
- **Arrows**:
- `induction n` (recursive case)
- `cases n` (pattern matching)
- `exfaslo` (failure path)
- **Flow**:
- Starts with `n : N`, branches into cases, and terminates in `false` or proceeds to `ih`.
- **Right Section (Tactic Generation)**:
- **Boxes**:
- `n’ : N` (new variable)
- `ih: add 0 n’ = n’` (inductive hypothesis)
- `add 0 (n’+1) = n’+1` (inductive step)
- **Arrows**:
- `simp [add, ih]` (simplification tactic)
- `rw [ih]` (rewrite tactic)
- `linarith` (linear arithmetic tactic)
- **Flow**:
- Connects to a **language model** (network of nodes) and **formal math library** (stacked cylinders).
- **Visual Elements**:
- **Orange Boxes**: Highlight key steps (`simp [add, ih]`, `rw [ih]`, `linarith`).
- **Green Checkmark**: Indicates successful proof completion.
- **Dashed Arrows**: Represent connections between the language model and tactic suggestions.
### Detailed Analysis
- **Proof Search**:
- The left side models a recursive proof structure for `add 0 n = n`.
- Base case: `add 0 0 = 0`.
- Inductive step: `add 0 (n+1) = n+1` (using `ih: add 0 n = n`).
- Failure path: `exfaslo` leads to `false`.
- **Tactic Generation**:
- The right side uses a **language model** (neural network) to generate tactics based on the local context (`n’ : N`, `ih`).
- Tactics include:
- `simp [add, ih]`: Simplify using `add` and `ih`.
- `rw [ih]`: Rewrite using the inductive hypothesis.
- `linarith`: Solve linear arithmetic constraints.
- The **formal math library** provides foundational axioms (e.g., `add 0 0 = 0`).
### Key Observations
- The **green checkmark** signifies a valid proof path via `simp [add, ih]`.
- **Orange boxes** emphasize critical steps in the tactic generation process.
- The **language model** acts as a bridge between formal logic and heuristic tactics.
### Interpretation
This diagram demonstrates a hybrid approach to automated theorem proving:
1. **Formal Logic**: The left side enforces strict inductive reasoning for `add 0 n = n`.
2. **Machine Learning**: The right side uses a language model to suggest tactics (e.g., `simp`, `rw`, `linarith`) based on the local context.
3. **Integration**: The formal math library grounds the proof in axiomatic rules, while the language model introduces flexibility.
The workflow highlights the interplay between rigorous formal methods and heuristic search, enabling efficient proof discovery. The green checkmark confirms that the system can successfully validate proofs, while the orange boxes underscore the importance of tactic selection in complex cases.
