\n ## Diagram: Proof State Transition ### Overview The image depicts a diagram illustrating the state transitions during a formal proof, likely within a proof assistant system. It shows how tactics are applied to evolve the proof state from an initial condition to a completed proof. The diagram is structured with a "Tactics" column on the left and corresponding "States" column on the right, connected by arrows indicating the application of tactics. ### Components/Axes The diagram consists of the following components: * **Theorem Statement:** `theorem my_subset_insec_mono (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by` * **Tactics Column:** Labeled "Tactics" on the left side. * **States Column:** Labeled "States" on the right side. * **Arrows:** Indicate the application of tactics and the resulting state transitions. * **Initial State:** Labeled "Initial state" * **Proof Finished:** Indicates the final state of the proof. ### Detailed Analysis or Content Details **Theorem Statement:** The theorem states: "For all sets s and t, if s is a subset of t, then the intersection of s and u is a subset of the intersection of t and u." The notation `s ⊆ t` represents "s is a subset of t", and `∩` represents the intersection operation. **Initial State:** * `h : s ⊆ t` - Hypothesis: s is a subset of t. * `⊢ s ∩ u ⊆ t ∩ u` - Goal: Prove that s intersection u is a subset of t intersection u. **First Tactic & State Transition:** * **Tactic:** `intro x xsu` - Introduces a variable `x` and assumes `x ∈ s ∩ u`. * **State:** * `h : s ⊆ t` - Hypothesis remains unchanged. * `xsu : x ∈ s ∩ u` - New assumption: x is an element of the intersection of s and u. * `⊢ x ∈ t ∩ u` - New goal: Prove that x is an element of the intersection of t and u. **Second Tactic & State Transition:** * **Tactic:** `exact (h xsu.1, xsu.2)` - Applies the hypothesis `h` and the components of `xsu` to directly prove the goal. * **State:** * `Proof Finished` - The proof is complete. ### Key Observations The diagram demonstrates a simple proof by introduction and application of a hypothesis. The tactic `intro` introduces a new variable and assumption, while `exact` uses existing assumptions and the goal to complete the proof. The diagram clearly shows the evolution of the proof state with each tactic application. ### Interpretation This diagram illustrates a fundamental proof technique in formal verification and theorem proving. The use of tactics allows for a step-by-step construction of the proof, with each step transforming the proof state until the goal is reached. The diagram highlights the importance of clearly defining hypotheses and goals, and how tactics can be used to manipulate these elements to achieve a desired outcome. The `exact` tactic suggests a direct application of the hypothesis to the goal, indicating a relatively straightforward proof. The diagram is a visual representation of a logical argument, making it easier to understand the flow of the proof process.