\n ## Diagram: Lean Theorem Proving Workflow ### Overview The image presents a diagram illustrating a workflow within the Lean theorem proving system. It is divided into three main sections: "Proof tree", "Lean file", and "Project", demonstrating the progression from a proof strategy to its implementation in code and its associated project context. ### Components/Axes The diagram is structured into three distinct columns: 1. **Proof Tree (Left):** Visualizes the proof process with nodes representing goals, tactics, and intermediate states. 2. **Lean File (Center):** Displays the Lean code corresponding to the proof tree. Includes inductive definitions and theorem statements. 3. **Project (Right):** Shows GitHub repositories related to the Lean project, including contributor statistics (contributors, issues, stars, forks). ### Detailed Analysis or Content Details **1. Proof Tree:** * **Top Node:** `n : N` and `⊢ add 0 n = n` (Goal) * **Tactic:** `induction` * **Intermediate Node 1:** `⊢ add 0 0 = 0` and `n' : N` * **Tactic:** `rfl` * **Intermediate Node 2:** `⊢ add 0 (n'+1) = n'+1` * **Tactic:** `simp [add, ih]` **2. Lean File:** * `inductive Nat where` * `| zero : Nat` * `| succ : Nat → Nat` * `def add (m n : Nat) : Nat :=` * `match n with` * `| .zero => m` * `| .succ n' => .succ (add m n')` * `theorem add_zero (n : Nat) : add 0 n = n by` * `induction n with` * `| zero => rfl` * `| succ n ih => simp [add, ih]` **3. Project (GitHub Repositories):** * **Repository 1:** `leanprover-community/mathlib4` * Icon: Flag icons (representing multiple languages/countries) * Contributors: 300+ * Issues: 21 * Stars: 273 * Forks: 73 * Description: "The math library of Lean 4" * **Repository 2:** `teeth/pfr` * Icon: Bracketed angle symbol `<>` * Contributors: 29 * Issues: 58 * Stars: 103 * Forks: 38 * Description: "Repository for formalization of the Polynomial (Fermat library compatible and related result)" * **Repository 3:** `ImperialCollegeLondon/FLT` * Icon: Letter "I" * Contributors: 28 * Issues: 24 * Stars: 96 * Forks: 36 * Description: "Ongoing Lean formalization of the proof of Fermat's Last Theorem" ### Key Observations * The "Proof Tree" section demonstrates a simple inductive proof for the `add_zero` theorem. * The "Lean File" section provides the corresponding Lean code for the inductive definition of natural numbers (`Nat`) and the addition function (`add`). * The "Project" section highlights relevant GitHub repositories, indicating a collaborative effort in formalizing mathematical theorems using Lean. * The GitHub repositories show varying levels of activity, measured by contributors, issues, stars, and forks. `mathlib4` has significantly more contributors, stars, and forks than the other two repositories. ### Interpretation The diagram illustrates the core workflow of formal theorem proving using Lean. It shows how a mathematical proof is constructed logically (Proof Tree), translated into executable code (Lean File), and managed within a collaborative software development environment (Project). The connection between the proof tree and the Lean file is crucial: the tactics applied in the proof tree directly correspond to commands used in the Lean code. The GitHub repositories demonstrate the open-source nature of the Lean project and the community's involvement in building a comprehensive library of formalized mathematics. The varying statistics of the repositories suggest different levels of maturity and focus within the Lean ecosystem. The diagram effectively conveys the interplay between logical reasoning, code implementation, and collaborative development in the context of formal verification.