## Code Snippet: Lean Theorem Prover Code ### Overview The image shows a code snippet, likely from the Lean theorem prover, defining a lemma related to the inverse map of a Coxeter system. The code includes the lemma definition, simplification, unfolding, and application steps. ### Components/Axes * **Code Block:** The main component is a block of code with a dark background and syntax highlighting. * **Window Controls:** Three circles (red, yellow, green) are located at the top-left, resembling window controls. ### Detailed Analysis or ### Content Details The code snippet contains the following lines: 1. `lemma invmap.of_eq {S:Set G} [CoxeterSystem G S] {s :S} : invmap S s = s := by` 2. `simp [CoxeterSystem.Presentation.invmap]` 3. `unfold CoxeterSystem.toMatrix` 4. `apply CoxeterSystem.monoidLift.mapLift.of` ### Key Observations * The code defines a lemma named `invmap.of_eq`. * It involves sets `S` and `G`, and a Coxeter system. * The lemma states that `invmap S s = s`. * The proof uses simplification, unfolding, and application steps. ### Interpretation The code snippet is a formal proof within the Lean theorem prover. It defines and proves a property related to the inverse map in the context of Coxeter systems. The lemma `invmap.of_eq` likely establishes a fundamental relationship between an element `s` in a set `S` and its inverse map. The proof strategy involves simplifying the expression, unfolding definitions, and applying relevant theorems or functions. This type of code is used to formally verify mathematical statements and ensure their correctness.