## Screenshot: Lean Theorem Proof Interface ### Overview The image shows a code editor interface displaying a formal theorem proof in the Lean theorem prover. The editor uses a dark theme with syntax highlighting. The visible content includes a theorem declaration and an application of a built-in lemma. ### Components/Axes - **UI Elements**: - Top-left corner: Three circular buttons (red, yellow, green) for window management (close, minimize, maximize). - Dark-themed code editor with syntax highlighting. - **Code Structure**: - Theorem declaration: `theorem abs_add (x y : R) : |x + y| ≤ |x| + |y| := by` - Apply command: `apply abs_add_le` ### Detailed Analysis - **Theorem Declaration**: - Name: `abs_add` - Parameters: `x, y : R` (real numbers) - Statement: `|x + y| ≤ |x| + |y|` (triangle inequality) - Proof method: `by` (invokes Lean's proof assistant to complete the proof) - **Apply Command**: - Lemma used: `abs_add_le` (pre-existing lemma for the triangle inequality) ### Key Observations 1. The theorem explicitly states the triangle inequality for real numbers. 2. The `apply` command leverages Lean's built-in lemma `abs_add_le` to auto-generate the proof. 3. Syntax highlighting distinguishes keywords (`theorem`, `apply`), variables (`x`, `y`), and symbols (`|`, `:`, `≤`). ### Interpretation This screenshot demonstrates Lean's capability to formalize mathematical proofs. The theorem `abs_add` declares the triangle inequality, while `apply abs_add_le` uses Lean's type checker to automatically derive the proof from the pre-verified lemma `abs_add_le`. The structure reflects Lean's functional programming paradigm, where proofs are treated as programs. The use of `by` indicates reliance on Lean's automated proof search, showcasing the interplay between human-readable mathematics and machine-verified logic. No numerical data or trends are present; the focus is on formal syntax and proof structure.