## Code Snippet: Theorem Definitions ### Overview The image shows a code snippet, likely from a formal verification system or proof assistant, defining three mathematical theorems. The theorems are related to number theory and algebra, and they are proven using the `norm_num` tactic. ### Components/Axes The code snippet is displayed within a dark-themed window with standard macOS-style window controls (red, yellow, green circles) in the top-left corner. The text is formatted in a monospaced font. ### Detailed Analysis or ### Content Details The code defines three theorems: 1. **Theorem `mathd_numbertheory_254`**: * Statement: `(239 + 174 + 83) % 10 = 6` * Proof: `by norm_num` * Explanation: This theorem states that the remainder of the sum (239 + 174 + 83) divided by 10 is equal to 6. 2. **Theorem `mathd_numbertheory_342`**: * Statement: `54 % 6 = 0` * Proof: `by norm_num` * Explanation: This theorem states that the remainder of 54 divided by 6 is equal to 0. 3. **Theorem `mathd_algebra_304`**: * Statement: `91^2 = 8281` * Proof: `by norm_num` * Explanation: This theorem states that 91 squared is equal to 8281. The `norm_num` tactic is used to automatically prove these theorems. This tactic likely simplifies the expressions and performs the necessary calculations to verify the equality. ### Key Observations * All three theorems are proven using the same tactic: `norm_num`. * The theorems cover basic arithmetic and algebraic operations. * The naming convention for the theorems suggests they are part of a larger mathematical database or library. ### Interpretation The code snippet demonstrates the use of a formal verification system to define and prove mathematical theorems. The `norm_num` tactic simplifies the proof process by automating the necessary calculations. The theorems themselves are relatively simple, but they illustrate the basic principles of formal verification. The theorems are named in a way that suggests they are part of a larger collection, possibly used for automated reasoning or mathematical exploration.