## Screenshot: Code Snippet with Theorem Declaration ### Overview The image displays a code snippet from a formal proof assistant or theorem prover, likely written in a language like Lean, Coq, or Isabelle. The text is syntax-highlighted with color-coded elements, and the interface includes standard window control buttons (red, yellow, green circles) in the top-left corner. ### Components/Axes - **Window Controls**: Three circular buttons in the top-left corner: - Red (close) - Yellow (minimize) - Green (maximize) - **Text Content**: - **Line 1**: `theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by` - `theorem` (purple) - `Subset.trans` (red) - `r ⊆ s → s ⊆ t → r ⊆ t` (pink) - `:= by` (pink) - **Line 2**: `exact Set.Subset.trans` - `exact` (purple) - `Set.Subset.trans` (purple) ### Detailed Analysis - **Syntax Highlighting**: - Keywords (`theorem`, `exact`) are highlighted in purple. - Identifiers (`Subset.trans`, `Set.Subset.trans`) are in red. - Mathematical/logical symbols (`⊆`, `→`, `:=`, `by`) are in pink. - **Code Structure**: - The first line declares a theorem (`theorem`) named `Subset.trans`, which states that if `r` is a subset of `s` and `s` is a subset of `t`, then `r` is a subset of `t`. - The second line (`exact Set.Subset.trans`) likely references a built-in or previously defined transitive closure property for sets. ### Key Observations - The theorem asserts the transitivity of subset relations, a fundamental property in set theory. - The use of `by` suggests the proof is deferred to a tactic (e.g., `transitivity`) in the proof assistant. - The `exact` keyword indicates the theorem is being proven by directly applying a known lemma (`Set.Subset.trans`). ### Interpretation This snippet demonstrates a formal verification of the transitivity law for subset relations. The code is structured to: 1. **Declare a theorem** (`theorem Subset.trans`) with a hypothesis chain (`r ⊆ s → s ⊆ t → r ⊆ t`). 2. **Invoke a pre-existing lemma** (`Set.Subset.trans`) to complete the proof, avoiding manual derivation. The color coding enhances readability by distinguishing syntactic roles: - **Purple** for high-level constructs (`theorem`, `exact`). - **Red** for identifiers (theorem name, lemma reference). - **Pink** for logical/mathematical operators and proof tactics. The theorem’s validity relies on the correctness of the underlying proof assistant’s logic, ensuring that transitivity holds for all sets `r`, `s`, and `t`. This is critical in formal systems where such properties must be rigorously validated.