\n ## Diagram: Proof Tree for Commutative Property of Multiplication ### Overview This diagram depicts a proof tree exploring the validity of the commutative property of multiplication (a * b * c = b * a * c). It shows three branches stemming from the initial equation, each applying a different rewrite rule (rw) and leading to either a "ProofFinished" state or an "Invalid tactic" state. ### Components/Axes The diagram consists of rectangular nodes representing equations or states, connected by arrows indicating the application of a rewrite rule. Each arrow is labeled with the specific rewrite rule used. The final nodes are labeled "ProofFinished" or "Invalid tactic". ### Detailed Analysis or Content Details The diagram can be broken down into three branches: * **Branch 1 (Left):** * Starting Equation: `a * b * c = b * (a * c)` * Rewrite Rule: `rw [mul_comm a c]` * Intermediate Equation: `c * b * a = b * (a * c)` * Rewrite Rule: `rw [mul_comm b c]` * Final State: `Invalid tactic` * **Branch 2 (Center):** * Starting Equation: `a * b * c = b * (a * c)` * Rewrite Rule: `rw [mul_comm a b]` * Intermediate Equation: `b * a * c = b * (a * c)` * Rewrite Rule: `rw [mul_assoc]` * Intermediate Equation: `b * (a * c) = b * (a * c)` * Final State: `ProofFinished` * **Branch 3 (Right):** * Starting Equation: `a * b * c = b * (a * c)` * Rewrite Rule: `rw [mul_assoc]` * Intermediate Equation: `a * (b * c) = b * (a * c)` * Rewrite Rule: `rw [mul_comm b c]` * Intermediate Equation: `a * (c * b) = b * (a * c)` * Final State: `Invalid tactic` ### Key Observations The central branch successfully proves the commutative property using the `mul_comm a b` and `mul_assoc` rewrite rules. The other two branches lead to "Invalid tactic", indicating that the chosen rewrite rules do not lead to a valid proof. ### Interpretation This diagram demonstrates a formal verification process using rewrite rules to prove or disprove a mathematical property. The tree structure allows for exploring different proof paths. The "ProofFinished" node in the center branch confirms that the commutative property of multiplication can be proven using the specified rewrite rules. The "Invalid tactic" nodes highlight that not all rewrite rule applications lead to a valid proof, emphasizing the importance of choosing the correct sequence of rules. The diagram illustrates a specific instance of automated theorem proving, where a system attempts to find a sequence of valid transformations to reach a desired conclusion. The rewrite rules `mul_comm` and `mul_assoc` represent fundamental properties of multiplication. The diagram suggests that the order of applying these rules is crucial for achieving a successful proof.