\n ## Screenshot: Step Proof - Mathematical Proof Assistant ### Overview This is a screenshot of a mathematical proof assistant application, likely designed to help users construct formal proofs. The interface displays a problem statement, a series of proof steps, and buttons to manipulate the proof. The application appears to be in the process of proving a statement about integers and divisibility. ### Components/Axes The screenshot contains the following key elements: * **Title Bar:** "Step Proof" * **Menu Bar:** Contains "File" option. * **Problem Statement Area:** A large text box displaying the problem to be proven. * **Proof Steps Area:** A series of boxes, each representing a step in the proof. Each step has buttons: "HIDE", "REGEN", "HOLD", "UNDO". * **Status Bar:** Displays "Status: Proof has been completed." * **"Add Proof" Button:** Allows the user to add a new proof step. ### Content Details The problem statement is: "Let $a$, $b$, $n$ be integers. Prove that if $a \mid n$ and $b \mid n$ with $gcd(a, b) = 1$ then $ab \mid n$." The proof steps are as follows: 1. "Since $a \mid n$, $n$ could be rewritten as $n = a(n/a)*a$." have h1: "n = n div a * a" using assms(1) sledgehammer 2. "Therefore, $b \mid n$ means $b \mid (n/a)*a$." have h2: "b dvd n div a * a" using assms(2) sledgehammer 3. "Since $gcd(a, b) = 1$ and $b \mid (n/a)*a$, it means that $b \mid (n/a)$." have h3: "b dvd n div a" using assms(3) h2 sledgehammer 4. "By multiple $a$ on the both side of $b \mid (n/a)$, we will know that $b*a \mid (n/a)*a$ and we can get $a*b \mid n$." have h4: "a * b dvd n" using h3 sledgehammer 5. "then show ?thesis by auto" 6. "QED" ### Key Observations The proof appears to be complete, as indicated by the status bar. The proof steps utilize mathematical notation (e.g., "$a \mid n$", "$gcd(a, b)$") and refer to "assms" and "sledgehammer," suggesting the use of automated theorem proving techniques. The buttons "HIDE", "REGEN", "HOLD", and "UNDO" suggest the user can manipulate the proof steps. ### Interpretation The screenshot demonstrates a mathematical proof assistant in action. The application guides the user through the process of constructing a formal proof, providing tools for both manual input and automated reasoning. The proof itself concerns a fundamental property of divisibility in integers: if two numbers divide a common multiple and are relatively prime, then their product also divides the multiple. The use of "sledgehammer" suggests the system is attempting to automatically apply known theorems or lemmas to simplify the proof. The completed status indicates the system has successfully verified the proof. The interface is designed for a user familiar with mathematical logic and proof techniques. The application is likely used for educational purposes or for formal verification of mathematical statements.