\n ## Screenshot: Mathematical Theorem Proof ### Overview The image is a screenshot of a terminal window displaying a mathematical theorem and its proof, likely written in a formal proof assistant language such as Lean. The background is dark gray, and the text is white. The top-left corner shows three colored circles (red, yellow, green), likely indicating the operating system is macOS. ### Components/Axes There are no axes or traditional chart components. The screenshot consists entirely of text. ### Detailed Analysis or Content Details The text content is as follows: ``` theorem induction_12dvd4expnp1p20 (n : ℕ) : 12 | 4^(n+1) + 20 := by norm_num induction' n with hn simp omega ``` Breaking down the content: * `theorem induction_12dvd4expnp1p20 (n : ℕ) : 12 | 4^(n+1) + 20 := by`: This line declares a theorem named `induction_12dvd4expnp1p20`. It takes a natural number `n` as input (`n : ℕ`) and asserts that 12 divides `4^(n+1) + 20` (denoted by `12 | 4^(n+1) + 20`). The `:= by` indicates the start of the proof. * `norm_num`: This is a tactic, likely a command to normalize numerical expressions. * `induction' n with hn`: This is an induction tactic. It performs mathematical induction on the variable `n`. `hn` is the induction hypothesis. * `simp`: This is a simplification tactic, likely applying simplification rules to the current goal. * `omega`: This is a tactic that attempts to automatically complete the proof using various decision procedures. ### Key Observations The code snippet represents a formal proof of a divisibility theorem. The theorem states that for all natural numbers `n`, 12 divides `4^(n+1) + 20`. The proof uses standard tactics like normalization, induction, simplification, and an automated proof search tactic (`omega`). ### Interpretation The screenshot demonstrates a formal mathematical proof being constructed within a proof assistant environment. The theorem itself suggests a relationship between powers of 4 and divisibility by 12. The use of tactics indicates a step-by-step approach to constructing a logically sound argument. The `omega` tactic suggests that the proof assistant was able to automatically complete the proof after the initial steps. This is a common workflow in formal verification, where a human provides the initial strategy, and the machine handles the detailed reasoning. The theorem is likely a simple example used for demonstration or educational purposes.