## Diagram: Proof State Transition ### Overview The image depicts a proof state transition diagram, showing the steps involved in proving a theorem related to subsets and intersections. It illustrates the initial state, the tactics applied, the resulting states, and the final "Proof Finished" state. ### Components/Axes * **Theorem Statement:** `theorem my_subset_intsec_mono (h : s⊆t) : s ∩ u ⊆ t ∩ u := by` * This defines the theorem being proven: Given that set `s` is a subset of set `t` (denoted `s⊆t`), then the intersection of `s` and `u` is a subset of the intersection of `t` and `u` (denoted `s ∩ u ⊆ t ∩ u`). * **Initial State:** * `h : s ⊆ t` (Hypothesis: `s` is a subset of `t`) * `⊢ s ∩ u ⊆ t ∩ u` (Goal: Prove that `s ∩ u` is a subset of `t ∩ u`) * **Tactics:** * `intro x xsu` (Introduce an arbitrary element `x` and assume it belongs to `s ∩ u`) * `exact <h xsu.1, xsu.2>` (Apply the hypothesis `h` and the assumptions about `x` to complete the proof) * **States:** * `h : s ⊆ t` (Hypothesis remains) * `xsu : x ∈ s ∩ u` (Assumption: `x` belongs to the intersection of `s` and `u`) * `⊢ x ∈ t ∩ u` (New Goal: Prove that `x` belongs to the intersection of `t` and `u`) * **Final State:** `Proof Finished` * **Arrows:** Two downward-pointing arrows indicate the transitions between states. * **Braces with labels:** * A brace on the right side of the initial state is labeled "Initial state". * A brace on the left side of the tactics is labeled "Tactics". * A brace on the right side of the intermediate states is labeled "States". ### Detailed Analysis or Content Details The diagram shows a step-by-step transformation of the proof state. 1. **Initial State:** The proof starts with the hypothesis `h : s ⊆ t` and the goal `⊢ s ∩ u ⊆ t ∩ u`. 2. **Tactic Application:** The tactic `intro x xsu` introduces an element `x` and assumes it belongs to `s ∩ u`. This means `xsu : x ∈ s ∩ u`. 3. **Intermediate State:** The goal is now to prove that `x ∈ t ∩ u`. 4. **Tactic Application:** The tactic `exact <h xsu.1, xsu.2>` uses the hypothesis `h` (that `s ⊆ t`) and the fact that `x ∈ s ∩ u` to deduce that `x ∈ t ∩ u`. 5. **Final State:** The goal is achieved, and the proof is finished. ### Key Observations * The diagram illustrates a simple proof strategy using introduction and application of a hypothesis. * The tactics directly address the goal of the proof. * The diagram clearly shows the transition from the initial state to the final state through the application of tactics. ### Interpretation The diagram demonstrates a formal proof of the theorem `my_subset_intsec_mono`. It shows how, given that `s` is a subset of `t`, proving that `s ∩ u` is a subset of `t ∩ u` can be achieved by introducing an arbitrary element `x` belonging to `s ∩ u` and then showing that `x` must also belong to `t ∩ u`. The tactic `exact <h xsu.1, xsu.2>` represents the logical step where the subset relationship `s ⊆ t` is used to infer that if `x` is in `s`, it must also be in `t`. This, combined with the fact that `x` is in `u`, proves that `x` is in `t ∩ u`, thus completing the proof.