\n ## Code Snippet: Lean Theorem Proof ### Overview The image is a screenshot of a code editor window (likely on macOS, indicated by the three colored window control dots in the top-left corner) displaying a theorem and its proof written in the Lean theorem prover language. The code is presented with syntax highlighting on a dark background. ### Components/Axes This is not a chart or diagram with axes. The components are the lines of code within the editor window. **Window Elements:** * **Top-left corner:** Three circular window control buttons (red, yellow, green). * **Main content area:** A block of code with syntax highlighting. The text colors are part of the theme and help distinguish language elements. ### Detailed Analysis / Content Details The following text is present in the image, transcribed exactly as it appears, including syntax highlighting colors noted descriptively: 1. **Line 1 (Pink text):** `theorem mathd_algebra_270` * This declares a theorem named `mathd_algebra_270`. 2. **Line 2 (Mixed colors):** ` (f : ℝ → ℝ)` * This is a parameter declaration. It states that `f` is a function from the real numbers (`ℝ`) to the real numbers (`ℝ`). The symbol `ℝ` is rendered in a distinct color (likely blue or cyan). 3. **Line 3 (Mixed colors):** ` (h₀ : ∀ x, x ≠ -2 → f x = 1 / (x + 2)) :` * This declares a hypothesis named `h₀`. * The hypothesis states: "For all `x`, if `x` is not equal to `-2`, then `f x` equals `1 / (x + 2)`." * Symbols: `∀` (for all), `≠` (not equal), `→` (implies), `ℝ` (real numbers) are present. 4. **Line 4 (Mixed colors):** ` f (f 1) = 3/7 := by` * This is the goal of the theorem: to prove that `f (f 1)` equals `3/7`. * The `:= by` indicates the start of the proof. 5. **Line 5 (Mixed colors):** ` set_option tactic.skipAssignedInstances false in norm_num [h₀]` * This is the proof tactic. * It first sets an option (`tactic.skipAssignedInstances`) to `false`. * Then it uses the `norm_num` tactic, providing the hypothesis `h₀` as a parameter in square brackets `[h₀]`. ### Key Observations * **Language:** The code is written in Lean, a formal proof management system and functional programming language. * **Structure:** The code follows a standard Lean theorem structure: `theorem name : statement := by proof`. * **Mathematical Content:** The theorem defines a function `f` with a specific rational expression and asks to compute the value of its composition at a point (`f(f(1))`). * **Proof Method:** The proof relies on the `norm_num` tactic, which is used for normalizing numerical expressions, and it explicitly uses the given hypothesis `h₀`. ### Interpretation This image captures a formal mathematical statement and its verification within a computer-assisted proof system. * **What the data suggests:** The code represents a solved problem, likely from a mathematics competition or textbook (suggested by the name `mathd_algebra_270`). It demonstrates the use of a theorem prover to verify an algebraic computation. * **How elements relate:** The hypothesis `h₀` defines the function `f`. The goal is a specific numerical claim about `f`. The proof tactic `norm_num` uses the definition in `h₀` to compute the value `f(1) = 1/(1+2) = 1/3`, and then `f(f(1)) = f(1/3) = 1/(1/3 + 2) = 1/(7/3) = 3/7`, thereby proving the goal. * **Notable aspects:** The use of `set_option tactic.skipAssignedInstances false` is a technical detail specific to the Lean tactic state, indicating a precise control over how the `norm_num` tactic behaves in this context. The entire snippet is a self-contained, machine-checkable proof of a simple algebraic identity.