## Diagram: Coq Theorem Proof Structure ### Overview The image depicts a formal proof structure for a Coq theorem named `my_subset_intsec_mono`. It illustrates the logical flow from the theorem's hypothesis to its conclusion using Coq tactics and state transitions. The diagram is divided into two main sections: **Tactics** (proof steps) and **States** (logical assertions at each stage). ### Components/Axes - **Theorem Statement**: `theorem my_subset_intsec_mono (h : s ⊆ t) : s ∪ u ⊆ t ∪ u := by` - **Hypothesis**: `h : s ⊆ t` (s is a subset of t). - **Conclusion**: `s ∪ u ⊆ t ∪ u` (the union of s and u is a subset of the union of t and u). - **Tactics Section**: - **Step 1**: `intro x xsu` - Introduces variables `x` and `xsu` (likely representing elements or subsets). - **Step 2**: `exact (h xsu.1, xsu.2)` - Applies the hypothesis `h` to the elements `xsu.1` and `xsu.2` to complete the proof. - **States Section**: - **Initial State**: - `h : s ⊆ t` (hypothesis). - Goal: `s ∪ u ⊆ t ∪ u`. - **Final State**: - `Proof Finished` (conclusion achieved). ### Detailed Analysis - **Theorem Logic**: The theorem asserts that if `s` is a subset of `t`, then adding `u` to both sides preserves the subset relationship. This is a standard result in set theory, leveraging the transitivity of subset inclusion. - **Tactics Breakdown**: 1. `intro x xsu`: Introduces variables to represent elements or subsets involved in the proof. 2. `exact (h xsu.1, xsu.2)`: Directly applies the hypothesis `h` to specific elements (`xsu.1` and `xsu.2`) to satisfy the goal. - **State Transitions**: - The initial state includes the hypothesis `h : s ⊆ t` and the goal `s ∪ u ⊆ t ∪ u`. - After applying the tactics, the proof concludes with `Proof Finished`, indicating the goal is satisfied. ### Key Observations - The proof relies on the hypothesis `h : s ⊆ t` to establish the subset relationship for the unions. - The use of `exact` suggests the proof is straightforward, requiring no additional lemmas or complex reasoning. - The diagram emphasizes the modularity of Coq proofs, separating tactics (steps) from states (logical assertions). ### Interpretation This diagram demonstrates how Coq formalizes mathematical proofs by explicitly tracking hypotheses, goals, and proof steps. The theorem `my_subset_intsec_mono` is a foundational result in set theory, and the proof structure highlights Coq's ability to automate simple proofs using built-in tactics. The absence of intermediate states suggests the proof is concise, relying on direct application of the hypothesis. The diagram underscores the importance of clear hypothesis management in formal verification systems like Coq.