## Code Snippet: Lean 4 Lemma Definition
### Overview
The image is a screenshot of a code editor or terminal window displaying a single lemma written in the Lean 4 proof assistant language. The code is presented with syntax highlighting on a dark background. The content is a formal mathematical statement and its proof regarding an `invmap` function within the context of a `CoxeterSystem`.
### Components
1. **Window Frame**: A dark gray, rounded rectangular window. In the top-left corner are three colored circles (red, yellow, green), characteristic of macOS window controls.
2. **Code Area**: The main content area has a dark charcoal background (`#2d2d2d` approximate). The text is monospaced and uses syntax highlighting with the following approximate color scheme:
* Keywords (`lemma`, `by`, `simp`, `unfold`, `apply`): Light pink/salmon.
* Type/Class names (`Set`, `CoxeterSystem`): Light green.
* Function/Constructor names (`invmap.of_eq`, `Presentation.invmap`, `toMatrix`, `monoidLift.mapLift.of`): Light blue/cyan.
* Variables and symbols (`S`, `G`, `s`): White or light gray.
* Punctuation and operators (`{`, `}`, `[`, `]`, `:`, `=`, `.`): White or light gray.
### Content Details (Transcription)
The complete textual content of the code snippet is transcribed below. The language is **English**, used for the Lean 4 syntax and mathematical terms.
```lean
lemma invmap.of_eq {S:Set G} [CoxeterSystem G S] {s :S} : invmap S s = s := by
simp [CoxeterSystem.Presentation.invmap]
unfold CoxeterSystem.toMatrix
apply CoxeterSystem.monoidLift.mapLift.of
```
**Line-by-Line Breakdown:**
* **Line 1 (Lemma Declaration):** `lemma invmap.of_eq {S:Set G} [CoxeterSystem G S] {s :S} : invmap S s = s := by`
* Declares a lemma named `invmap.of_eq`.
* Implicit parameters: `S` is a `Set` of `G`; there is an instance of a `CoxeterSystem` for `G` and `S`; `s` is an element of `S`.
* The statement to prove: `invmap S s = s`.
* The proof begins with `by`.
* **Line 2 (Proof Step):** ` simp [CoxeterSystem.Presentation.invmap]`
* Uses the `simp` tactic, simplifying the goal using the definition or lemma `CoxeterSystem.Presentation.invmap`.
* **Line 3 (Proof Step):** ` unfold CoxeterSystem.toMatrix`
* Unfolds (replaces with its definition) the constant `CoxeterSystem.toMatrix`.
* **Line 4 (Proof Step):** ` apply CoxeterSystem.monoidLift.mapLift.of`
* Applies the theorem or lemma `CoxeterSystem.monoidLift.mapLift.of` to close the goal.
### Key Observations
1. **Formal Mathematics**: The code is a formal proof script in Lean 4, a dependently typed functional programming language and proof assistant.
2. **Subject Matter**: The lemma pertains to **Coxeter groups** or **Coxeter systems**, a fundamental structure in geometric group theory and algebraic combinatorics. The `invmap` likely refers to an inversion map.
3. **Proof Structure**: The proof is concise (3 tactics), suggesting the result follows directly from definitions and previously established lemmas within the formalized theory.
4. **Visual Presentation**: The syntax highlighting aids readability by distinguishing language keywords, types, and user-defined terms. The dark theme is common in modern development environments.
### Interpretation
This image captures a moment in the formal verification of mathematical theorems. The lemma `invmap.of_eq` asserts that for an element `s` in the generating set `S` of a Coxeter system, the inversion map (`invmap`) acts as the identity. This is a foundational property, as inversion is a key operation in Coxeter group theory.
The proof strategy reveals how such properties are established in a proof assistant:
1. **Simplification (`simp`)**: Leans on the core definition of `invmap` provided in the `Presentation` module.
2. **Unfolding (`unfold`)**: Exposes the underlying matrix representation of the Coxeter system (`toMatrix`), which is central to its definition.
3. **Application (`apply`)**: Uses a more general lifting property (`monoidLift.mapLift.of`) about homomorphisms from the monoid generated by `S` to the group `G`.
The snippet demonstrates the **hierarchical and modular nature** of formalized mathematics. The proof doesn't re-derive basic facts but instead chains together existing components (`Presentation.invmap`, `toMatrix`, `monoidLift`). This ensures consistency and allows complex theorems to be built from verified, reusable blocks. The existence of this lemma in a codebase implies an ongoing effort to formalize the theory of Coxeter systems, which has applications in areas like reflection groups, Lie theory, and combinatorics.
