## Code Snippet: Lean 4 Theorem Proof
### Overview
The image is a screenshot of a code editor or terminal window with a dark theme, displaying a formal mathematical theorem and its proof written in the Lean 4 programming language and theorem prover. The content is a self-contained proof of a logarithmic identity.
### Components / Visual Elements
1. **Window Frame**: A dark gray rectangular window with rounded corners, centered on a light gray background.
2. **Window Controls**: In the top-left corner of the window, there are three colored circles arranged horizontally:
* Left: Red
* Center: Yellow/Orange
* Right: Green
These are standard macOS-style window control buttons (close, minimize, maximize).
3. **Code Content**: The main body of the window contains four lines of monospaced text with syntax highlighting. The text is left-aligned with a consistent indentation for the proof tactics.
### Detailed Content (Transcription)
The following text is present in the code block. The syntax highlighting colors are described in brackets.
**Line 1:**
`theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by`
* `theorem` (Yellow/Gold)
* `mathd_algebra_484` (White/Light Gray)
* `:` (White/Light Gray)
* `Real.log` (Purple/Magenta)
* `27` (Cyan/Teal)
* `/` (White/Light Gray)
* `Real.log` (Purple/Magenta)
* `3` (Cyan/Teal)
* `=` (White/Light Gray)
* `3` (Cyan/Teal)
* `:=` (White/Light Gray)
* `by` (Yellow/Gold)
**Line 2 (Indented):**
` field_simp`
* `field_simp` (Yellow/Gold)
**Line 3 (Indented):**
` rw [← Real.log_rpow]`
* `rw` (Yellow/Gold)
* `[` (White/Light Gray)
* `←` (White/Light Gray)
* `Real.log_rpow` (Purple/Magenta)
* `]` (White/Light Gray)
**Line 4 (Indented):**
` all_goals norm_num`
* `all_goals` (Yellow/Gold)
* `norm_num` (Yellow/Gold)
### Key Observations
* **Language & Context**: The code is written in Lean 4, a functional programming language and interactive theorem prover used for formal verification of mathematics and software.
* **Theorem Statement**: The theorem, named `mathd_algebra_484`, asserts the mathematical identity: the logarithm of 27 to the base *e* (natural log) divided by the logarithm of 3 to the base *e* equals 3. This is a specific instance of the change-of-base formula and the property `log_b(b^k) = k`, since 27 = 3³.
* **Proof Structure**: The proof is concise and uses standard Lean tactics:
1. `field_simp`: A tactic to simplify expressions in a field (like real numbers), likely clearing denominators or simplifying the fraction.
2. `rw [← Real.log_rpow]`: Rewrites the expression using the theorem `Real.log_rpow` in the reverse direction (`←`). `Real.log_rpow` likely states that `log(x^r) = r * log(x)` for real `x` and `r`. Applying it in reverse would transform `log(27)` into `log(3^3)` and then to `3 * log(3)`, allowing cancellation with the denominator.
3. `all_goals norm_num`: Applies the `norm_num` tactic to all remaining goals. `norm_num` is a tactic that solves numerical goals by normalizing arithmetic expressions.
* **Visual Design**: The dark theme with distinct syntax highlighting (yellow for keywords, purple for constants/functions, cyan for numbers) is designed for code readability. The window styling is minimalist.
### Interpretation
This image captures a moment of formal mathematical verification. It demonstrates how modern theorem-proving assistants like Lean are used to encode and rigorously verify mathematical truths.
* **What the data suggests**: The code successfully proves a basic algebraic identity. The choice of tactics (`field_simp`, `rw`, `norm_num`) indicates the proof leverages Lean's library of simplification rules and arithmetic normalization, showing the power of automated reasoning within a formal system.
* **How elements relate**: The theorem statement defines the goal. The `by` keyword introduces the proof block. Each indented line is a tactic that transforms the proof state. The sequence of tactics logically reduces the initial goal to a trivial arithmetic truth that `norm_num` can discharge.
* **Notable aspects**: The theorem name `mathd_algebra_484` suggests it may be part of a larger dataset or challenge (like the Mathlib library or a benchmarking suite). The proof's brevity highlights the effectiveness of Lean's tactic framework for algebraic manipulation. There are no outliers or anomalies; this is a correct and complete formal proof of the stated identity. The image serves as a clear example of human-readable, machine-checked mathematics.
