## 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.