## Diagram: Proof Tree, Lean File, and GitHub Project
### Overview
The image presents a diagram that connects a proof tree, a Lean file snippet, and a GitHub project, illustrating the relationship between mathematical proofs, their formalization in the Lean theorem prover, and their availability on GitHub.
### Components/Axes
* **Proof Tree (Left):**
* A tree-like structure representing a mathematical proof.
* Nodes contain mathematical statements and tactics.
* "Local context" and "Goal" are indicated.
* Checkmark icons indicate successful proof steps.
* **Lean File (Center):**
* A code snippet from a Lean file.
* Defines natural numbers and addition.
* Includes a theorem and its proof.
* **Project (Right):**
* Represents a GitHub project.
* Shows project names, descriptions, and statistics (contributors, issues, stars, forks).
* Includes the GitHub logo.
### Detailed Analysis
**Proof Tree (Left):**
* **Top Node:**
* `n : N` (n is a natural number)
* `⊢ add 0 n = n` (Goal: 0 + n = n)
* **Tactic Node:**
* `Tactic` label
* `induction n` (Induction on n)
* **Left Child Node:**
* `⊢ add 0 0 = 0` (0 + 0 = 0)
* `rfl` (Reflexivity tactic)
* Green checkmark below.
* **Right Child Node:**
* `n' : N` (n' is a natural number)
* `ih: add 0 n' = n'` (Inductive hypothesis: 0 + n' = n')
* `⊢ add 0 (n'+1) = n'+1` (Goal: 0 + (n'+1) = n'+1)
* `simp [add, ih]` (Simplify using addition and the inductive hypothesis)
* Green checkmark below.
**Lean File (Center):**
* `inductive Nat where`
* `| zero : Nat` (Zero is a natural number)
* `| succ : Nat → Nat` (Successor function from natural number to natural number)
* `def add (m n : Nat) : Nat :=` (Definition of addition)
* `match n with` (Pattern matching on n)
* `| .zero => m` (If n is zero, return m)
* `| .succ n' => .succ (add m n')` (If n is the successor of n', return the successor of m + n')
* `theorem add_zero (n : Nat) : add .zero n = n := by` (Theorem: 0 + n = n)
* `induction n with` (Induction on n)
* `| zero => rfl` (If n is zero, use reflexivity)
* `| succ n ih => simp [add, ih]` (If n is the successor of n', simplify using addition and the inductive hypothesis)
**Project (Right):**
* **Top Project:**
* `leanprover-community/mathlib4`
* "The math library of Lean 4"
* Contributors: ~305
* Issues: ~221
* Stars: ~1k
* Forks: ~273
* **Middle Project:**
* `teorth/pfr`
* "Repository for formalization of the Polynomial Freiman Ruzsa conjecture (and related results)"
* Contributors: ~29
* Issues: ~0
* Stars: ~123
* Forks: ~33
* Image of a person.
* **Bottom Project:**
* `ImperialCollegeLon/FLT`
* "Ongoing Lean formalisation of the proof of Fermat's Last Theorem"
* Contributors: ~23
* Issues: ~4
* Stars: ~166
* Forks: ~36
### Key Observations
* The diagram illustrates the process of formalizing mathematical proofs using the Lean theorem prover and making them available on GitHub.
* The proof tree shows the logical steps involved in proving a simple theorem.
* The Lean file snippet shows the code used to define natural numbers, addition, and the theorem itself.
* The GitHub project represents a repository where Lean code and proofs are stored and shared.
### Interpretation
The diagram demonstrates the workflow of formalizing mathematics using Lean and GitHub. It shows how a mathematical proof (proof tree) can be translated into a formal language (Lean file) and then shared with the community (GitHub project). This process allows for rigorous verification of mathematical results and promotes collaboration among mathematicians and computer scientists. The connection between the proof tree, the Lean code, and the GitHub repository highlights the importance of formalization in modern mathematics.
