## Code Snippet: Formal Theorem Statement in Lean 4 ### Overview The image displays a screenshot of a code editor or terminal window with a dark theme. It contains a single line of formal mathematical code, specifically a theorem statement and its proof in the Lean 4 theorem prover language. The theorem is the well-known "triangle inequality" for real numbers. ### Components/Visual Elements * **Window Frame:** A dark gray, rounded rectangle window is centered against a lighter gray gradient background. * **Window Controls:** In the top-left corner of the window, there are three circular buttons: red, yellow, and green, consistent with macOS window management controls. * **Text Content:** A single line of monospaced text is displayed in the main body of the window. The text is a valid Lean 4 code statement. ### Content Details **Primary Language:** The code syntax is **Lean 4**, a functional programming language and interactive theorem prover. The mathematical content is expressed in standard notation. **Transcription of Text:** ```lean theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |y| := by apply abs_add_le ``` **Breakdown of the Statement:** 1. **`theorem abs_add`**: Declares a theorem named `abs_add`. 2. **`(x y : ℝ)`**: Specifies the theorem's parameters. `x` and `y` are variables of type `ℝ` (the set of real numbers). 3. **`: |x + y| ≤ |x| + |y|`**: States the proposition to be proven. This is the triangle inequality: the absolute value of a sum is less than or equal to the sum of the absolute values. 4. **`:= by`**: Introduces the proof block. 5. **`apply abs_add_le`**: The proof tactic. It instructs the system to prove the goal by applying a previously proven lemma or theorem named `abs_add_le`, which is presumably a more general or foundational version of this inequality. ### Key Observations * **Formal Verification:** This is not a comment or documentation; it is executable, verifiable code within a proof assistant. The statement's truth is mechanically checked by the Lean compiler. * **Conciseness:** The proof is extremely concise, relying on a single tactic (`apply`) that references an existing result (`abs_add_le`). This suggests `abs_add_le` is a core lemma in the mathematical library being used. * **Syntax:** The code uses standard mathematical symbols (`ℝ`, `| |`, `≤`) integrated into the programming language's syntax. ### Interpretation This image captures a moment of formal mathematical verification. The data presented is not numerical but logical and syntactic. * **What it demonstrates:** It shows the formalization of a fundamental result from real analysis (the triangle inequality) in a computer-checked format. The theorem `abs_add` is proven by appealing to a more basic lemma, `abs_add_le`, illustrating the hierarchical structure of mathematical knowledge in such systems. * **How elements relate:** The window and its controls are merely the container. The core information is the single line of code, which itself has a clear structure: name, parameters, statement, and proof method. The proof method (`apply abs_add_le`) directly links this specific theorem to a broader mathematical framework. * **Notable aspects:** The primary "anomaly" for a general viewer is that this is a *proof*, not just a statement. The `:= by apply...` portion is the critical component that transforms a mathematical claim into a verified fact within the system. The image contains no charts, graphs, or data tables; its entire informational value is contained in the precise textual code snippet.