## Code Snippet: Theorem Proving ### Overview The image shows a code snippet, likely from a theorem prover or interactive proof assistant. It defines a theorem related to real number logarithms and provides the steps to prove it. The code is displayed within a dark-themed window with standard close/minimize/maximize buttons. ### Components/Axes * **Window Controls:** Red, yellow, and green circles in the top-left corner, representing close, minimize, and maximize buttons respectively. * **Code Block:** * `theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by` * `field_simp` * `rw [← Real.log_rpow]` * `all_goals norm_num` ### Detailed Analysis or ### Content Details The code snippet defines a theorem named `mathd_algebra_484`. The theorem states that `Real.log 27 / Real.log 3` is equal to 3. The proof is provided using the `by` keyword, followed by a sequence of tactics: 1. `field_simp`: Simplifies the expression using field axioms. 2. `rw [← Real.log_rpow]`: Rewrites the expression using the `Real.log_rpow` theorem, applying it in reverse (indicated by the left arrow `←`). This theorem likely relates the logarithm of a power to the product of the exponent and the logarithm of the base. 3. `all_goals norm_num`: Normalizes numerical expressions in all remaining goals. ### Key Observations * The code uses a domain-specific language (DSL) for theorem proving. * The proof relies on simplification, rewriting, and normalization tactics. * The `Real.log_rpow` theorem is a key step in the proof. ### Interpretation The code snippet demonstrates a formal proof of a simple logarithmic identity. It showcases the use of automated tactics within a theorem prover to simplify and solve mathematical problems. The theorem `Real.log_rpow` is likely used to transform `Real.log 27` into `Real.log (3^3)`, which then simplifies to `3 * Real.log 3`, allowing for cancellation with the denominator. The `norm_num` tactic likely handles the final numerical evaluation.

\n ## Screenshot: Code Snippet - Mathematical Theorem ### Overview The image is a screenshot of a terminal or code editor window displaying a mathematical theorem and its proof steps, likely from a formal proof assistant system. The background is a dark grey, and the text is white. There are three colored dots in the top-left corner: red, yellow, and green. ### Components/Axes There are no axes or traditional chart components. The key elements are the text lines representing the theorem and proof commands. The colored dots appear to be status indicators. ### Detailed Analysis or Content Details The text content is as follows: ``` theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by field simp rw [← Real.log_rpow] all_goals norm_num ``` * **theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by**: This line declares a theorem named `mathd_algebra_484`. It states that the logarithm base 3 of 27 divided by the logarithm base 3 of 3 is equal to 3. The `:= by` indicates the start of the proof. * **field simp**: This is a proof command, likely instructing the system to simplify the expression using field arithmetic rules. * **rw [← Real.log_rpow]**: This is a rewrite rule command. It applies the rule `Real.log_rpow` to the expression. The `←` suggests that the rule is applied in reverse. `Real.log_rpow` likely relates to the logarithm of a power. * **all_goals norm_num**: This command instructs the system to normalize the numerical values in all proof goals. ### Key Observations The theorem is a basic logarithmic identity. The proof appears to be concise and uses standard proof tactics for a formal proof assistant. The colored dots in the top-left corner likely indicate the status of the proof process (e.g., red for error, yellow for warning, green for success). ### Interpretation The screenshot demonstrates a formal proof of a simple mathematical identity using a proof assistant. The commands suggest a tactic-based proof style, where the system automatically applies simplification and rewriting rules to reach the desired conclusion. The theorem itself is a straightforward application of logarithmic properties. The use of a formal proof assistant ensures the correctness and rigor of the proof. The dots are likely a visual indicator of the proof state, allowing the user to quickly assess the progress and identify any issues.

## 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.

## Screenshot: Code Snippet in Theorem Prover ### Overview The image shows a code snippet from a theorem prover (likely Lean or similar), displayed in a dark-themed code editor. The text is color-coded, with syntax highlighting for keywords, variables, and operations. The top-left corner contains three circular buttons (red, yellow, green), typical of macOS window controls. ### Components/Axes - **Top-left corner**: Three circular buttons (red, yellow, green) for window management. - **Code content**: - **Line 1**: `theorem mathd_algebra_484 : Real.log 27 / Real.log 3 = 3 := by` (yellow text). - **Line 2**: `field_simp` (yellow text). - **Line 3**: `rw [← Real.log_rpow]` (purple text). - **Line 4**: `all_goals norm_num` (yellow text). ### Detailed Analysis - **Line 1**: Declares a theorem named `mathd_algebra_484`, stating that `Real.log 27 / Real.log 3 = 3`. The `:= by` indicates the start of a proof. - **Line 2**: `field_simp` is a tactic to simplify expressions using field axioms. - **Line 3**: `rw [← Real.log_rpow]` applies a rewrite rule (`←` denotes leftward application) to `Real.log_rpow`, likely simplifying logarithmic expressions. - **Line 4**: `all_goals norm_num` normalizes all goals in the proof, possibly converting them to numerical form. ### Key Observations - The theorem proves that `log₃(27) = 3`, leveraging logarithmic identities (`log_b(a^c) = c log_b(a)`). - The use of `field_simp` and `rw` suggests the proof relies on algebraic simplification and rewrite rules. - `all_goals norm_num` implies the proof system automatically handles numerical normalization across all subgoals. ### Interpretation This code snippet demonstrates a formal proof in a theorem prover, verifying the logarithmic identity `log₃(27) = 3`. The `field_simp` and `rw` tactics automate algebraic manipulations, while `all_goals norm_num` ensures numerical consistency. The color coding (yellow for declarations, purple for tactics) aids readability, reflecting the structured nature of formal proofs. The theorem’s validity hinges on the correctness of the underlying mathematical axioms and the prover’s rewrite rules.