Ask AI: What is the Lean programming language?

Lean is a powerful proof assistant and programming language designed primarily for formal verification and theorem proving. Originally developed by Leonardo de Moura at Microsoft Research, Lean has gained significant attention in both academia and industry for its robust capabilities in formalizing mathematical proofs and ensuring software correctness. ### Key Features of Lean: 1. **Dependent Types**: Lean uses a dependent type theory, which allows types to depend on values. This enables highly expressive and precise type definitions. 2. **Interactive Proofs**: Lean provides an interactive environment where users can incrementally build and verify proofs. The Lean Prover IDE, often used within Visual Studio Code, supports this interactive development. 3. **Automation**: Lean includes various automatic tactics for assisting with proof construction, reducing the manual effort required. 4. **Libraries**: Lean has a growing mathematical library (mathlib) that contains a large corpus of formalized mathematics, making it a valuable resource for researchers and engineers. 5. **Functional Programming**: Lean supports functional programming paradigms, allowing users to write executable algorithms and reason about them formally within the same environment. ### Use Cases: 1. **Mathematical Theorems**: Lean is widely used for formalizing complex mathematical theories and proofs. 2. **Software Verification**: Lean can be used to formally verify software correctness, ensuring that programs behave as intended. 3. **Education**: Lean is also used as an educational tool for teaching formal logic, type theory, and proof techniques. ### Getting Started with Lean: To get started with Lean, you can follow these steps: 1. **Installation**: - Install Lean and its associated tools by following the instructions on the [Lean community website](https://leanprover.github.io/). 2. **Learning Resources**: - There are several tutorials and resources available to learn Lean, including the official documentation and community resources such as the "Theorem Proving in Lean" book. 3. **Join the Community**: - Engage with the Lean community through platforms like Zulip chat, GitHub, and forums to get support and collaborate on projects. ### Example: Here is a simple example of a proof in Lean: ```lean -- Define a theorem stating that for any natural number n, n + 0 = n theorem add_zero (n : nat) : n + 0 = n := nat.rec_on n (show 0 + 0 = 0, by refl) (assume n, assume ih : n + 0 = n, show (n + 1) + 0 = (n + 1), from calc (n + 1) + 0 = (n + 1) : by rw [nat.add_zero]) ``` This example demonstrates a basic theorem about natural numbers and uses Lean's interactive environment to build the proof step-by-step. Overall, Lean is a versatile and powerful tool for anyone interested in formal verification, theorem proving, or learning about formal methods in computer science and mathematics.