## Diagram: Proof Strategy for Associativity
### Overview
The image is a flowchart illustrating a proof strategy, likely in a formal verification or automated theorem proving context. It shows the steps taken to transform an initial expression using rewrite rules (rw) based on properties like commutativity (mul_comm) and associativity (mul_assoc) of multiplication. The diagram branches, indicating different possible paths in the proof, some leading to a successful "ProofFinished" state, while others lead to "Invalid tactic".
### Components/Axes
* **Nodes:** Rectangular boxes containing mathematical expressions. These represent states in the proof.
* **Edges:** Arrows connecting the nodes, indicating transitions between states.
* **Labels on Edges:** Text labels of the form "rw [rule_name]", indicating the rewrite rule applied to transition between states.
* **Terminal Nodes:** "ProofFinished" and "Invalid tactic" represent the end states of the proof attempts.
### Detailed Analysis
1. **Initial State (Top):**
* Expression: `a*b*c = b * (a* c)`
2. **First Level Transitions:**
* Left Branch: `rw [mul_comm a c]` leads to `c*b*a = b*(a*c)`
* Middle Branch: `rw [mul_comm a b]` leads to `b*a*c = b*(a*c)`
* Right Branch: `rw [mul_assoc]` leads to `a*(b*c) = b*(a*c)`
3. **Second Level Transitions:**
* From `b*a*c = b*(a*c)`: `rw [mul_assoc]` leads to `b*(a*c) = b*(a*c)`
* From `a*(b*c) = b*(a*c)`: `rw [mul_comm b c]` leads to `a*(c*b) = b*(a*c)`
4. **Third Level Transitions:**
* From `c*b*a = b*(a*c)`: `rw [mul_comm b c]` leads to "Invalid tactic"
* From `b*(a*c) = b*(a*c)`: leads to "ProofFinished"
* From `a*(c*b) = b*(a*c)`: `rw [mul_comm a b]` leads to "Invalid tactic"
### Key Observations
* The diagram shows a branching proof strategy.
* The rewrite rules used are `mul_comm` (commutativity of multiplication) and `mul_assoc` (associativity of multiplication).
* Not all paths lead to a successful proof. Some lead to "Invalid tactic", indicating a dead end.
* The central path, involving two applications of associativity, leads to "ProofFinished".
### Interpretation
The diagram illustrates a proof attempt to demonstrate or verify a property related to multiplication, likely associativity or a related identity. The branching structure shows different possible sequences of applying rewrite rules. The "Invalid tactic" outcomes suggest that some sequences of transformations do not lead to the desired result, highlighting the importance of choosing the correct proof strategy. The successful path demonstrates one way to achieve the proof. The diagram provides a visual representation of the proof process, making it easier to understand the steps involved and the reasoning behind them.
