\n ## Screenshot: Formal Mathematical Theorems in a Code Editor ### Overview The image is a screenshot of a dark-themed code editor or terminal window displaying two formal mathematical theorems and their proofs, written in a language consistent with a theorem prover like Lean. The window has a standard macOS-style title bar with three colored control buttons (red, yellow, green) in the top-left corner. The content consists of two distinct theorem statements, each followed by a brief proof script. ### Components/Axes * **Window Frame:** A dark gray rectangular window with rounded corners, centered on a light gray gradient background. * **Window Controls:** Three circular buttons in the top-left corner of the window. From left to right: red (close), yellow (minimize), green (maximize/zoom). * **Code Content Area:** The main dark area containing the text. The text uses syntax highlighting: * Keywords (`theorem`, `by`, `apply`, `cases`, `exact`) are in a salmon/pink color. * Theorem names (`my_lemma3`, `abs_lt`) are in a light blue/cyan color. * Mathematical symbols and variables are in white or light gray. * Specific symbols like `ε` (epsilon) and `ℝ` (real numbers) are highlighted in orange and purple, respectively. * Logical connectives (`→`, `↔`, `∧`) and operators (`<`, `*`) are in a teal color. ### Detailed Analysis The image contains two complete theorem statements and their proofs. **1. Theorem `my_lemma3`** * **Statement:** `∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε` * **Proof:** `:= by apply C03S01.my_lemma` * **Transcription & Translation:** The statement is in formal mathematical logic. It reads: "For all real numbers x, y, and epsilon, if epsilon is greater than 0, and epsilon is less than or equal to 1, and the absolute value of x is less than epsilon, and the absolute value of y is less than epsilon, then the absolute value of the product of x and y is less than epsilon." The proof applies a previously established lemma named `my_lemma` from a module or context `C03S01`. **2. Theorem `abs_lt`** * **Statement:** `|x| < y ↔ -y < x ∧ x < y` * **Proof:** `:= by cases x exact abs_lt` * **Transcription & Translation:** This is a fundamental property of absolute value. It states: "The absolute value of x is less than y if and only if negative y is less than x AND x is less than y." The proof uses a case analysis on `x` (likely considering `x ≥ 0` and `x < 0`) and then applies a previously defined fact or theorem named `abs_lt`. ### Key Observations * **Formal Verification Context:** The syntax (`theorem ... := by ...`, `apply`, `cases`, `exact`) is characteristic of interactive theorem provers used in formal mathematics and software verification. * **Proof Structure:** Both proofs are very concise, relying on applying existing lemmas (`C03S01.my_lemma`, `abs_lt`) rather than building the proof from basic axioms step-by-step. This suggests a hierarchical development where complex theorems are built upon simpler, previously proven ones. * **Mathematical Content:** The first theorem (`my_lemma3`) is a specific bound on the product of two small numbers, with the additional constraint that epsilon is at most 1. The second (`abs_lt`) is a standard, foundational equivalence used to remove absolute value signs in inequalities. * **Visual Layout:** The theorems are presented as two separate, clearly delineated blocks of code, each with its name, statement, and proof method. ### Interpretation This screenshot captures a moment in the process of formalizing mathematics using a computer-assisted proof system. The theorems are not just stated; they are *defined* in a language a machine can check for logical correctness. * **What the data suggests:** It demonstrates the use of a modular approach to building a mathematical library. Theorem `my_lemma3` depends on a lemma from a specific location (`C03S01`), indicating an organized codebase. The proof of `abs_lt` being used as a tactic name in its own proof is a curious detail that might indicate a naming convention or a recursive definition within the proof environment. * **How elements relate:** The two theorems are logically independent but thematically related, both dealing with inequalities and absolute values. They represent different levels of complexity: `abs_lt` is a basic tool, while `my_lemma3` is a more specific result that might be used in an analysis context (e.g., proving continuity or limits). * **Notable patterns/anomalies:** The most notable aspect is the extreme conciseness of the proofs. In a traditional written proof, these would require several lines of explanation. Here, the heavy lifting is done by the underlying proof framework and previously established facts, showcasing the power of formal verification systems to manage complex logical dependencies. The use of the same identifier `abs_lt` for both a theorem and a proof tactic is a minor but interesting syntactic feature of the specific proof language being used.