## 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.

\n ## Screenshot: Code Snippet - Lemma Definition ### Overview The image is a screenshot of a code editor displaying a lemma definition, likely from a formal proof assistant system (possibly Lean or Coq). The code appears to be related to Coxeter systems and involves concepts like inverse mapping, presentations, matrices, and monoid lifts. ### Components/Axes There are no axes or traditional chart components. The visible elements are: * **Colored Dots (Top-Left):** Three dots are present in the top-left corner: Red, Orange, and Green. These likely indicate status or selection within the editor. * **Code Block:** The main content is a block of code. * **Background:** A dark background with a subtle gradient. ### Detailed Analysis or Content Details The code block contains the following text: ``` 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.of ``` Let's break down the code: * `lemma invmap.of_eq {S:Set G} [CoxeterSystem G S] {s :S} : invmap S s = s := by`: This line defines a lemma named `invmap.of_eq`. It takes type parameters `S` (a set within `G`) and `G` itself, and a variable `s` belonging to the set `S`. The lemma states that `invmap S s` is equal to `s`. The `:= by` indicates the start of the proof. * `simp [CoxeterSystem.Presentation.invmap]`: This line instructs the proof assistant to simplify the expression using the `invmap` function from the `CoxeterSystem.Presentation` module. * `unfold CoxeterSystem.toMatrix`: This line instructs the proof assistant to expand the definition of `CoxeterSystem.toMatrix`. * `apply CoxeterSystem.monoidLift.of`: This line applies a theorem or lemma named `of` from the `CoxeterSystem.monoidLift` module. ### Key Observations * The code is syntactically correct (assuming the underlying system's syntax). * The lemma appears to be a fundamental property related to the inverse mapping within the context of Coxeter systems. * The proof strategy involves simplification and unfolding definitions, followed by applying a relevant theorem. ### Interpretation This code snippet represents a formal statement and its proof within a mathematical or computational system. The lemma `invmap.of_eq` likely establishes a crucial relationship between an element `s` and its inverse mapping `invmap S s` within a Coxeter system. The proof steps suggest that this relationship can be demonstrated by leveraging the system's definitions and existing theorems related to presentations, matrices, and monoid lifts. The use of a formal proof assistant ensures the rigor and correctness of the argument. The code is not presenting data in a visual way, but rather a logical statement and its justification.

## 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.

## Screenshot: Functional Programming Code Snippet ### Overview The image shows a code snippet from a functional programming or formal proof environment (likely Lean, Coq, or similar). The code defines a lemma (`invmap.of_eq`) about the inverse map of a Coxeter system, with a proof structured in three steps using `simp`, `unfold`, and `apply`. Syntax highlighting uses distinct colors for keywords, identifiers, and types. ### Components/Axes - **UI Elements**: - Top-left corner: Traffic light buttons (red, yellow, green) for window control. - **Code Structure**: - **Lemma Declaration**: - `lemma invmap.of_eq {S:Set G} [CoxeterSystem G S] {s : S} : invmap S s = s := by` - **Colors**: - `lemma` (red), `invmap.of_eq` (green), `{S:Set G}` (blue), `[CoxeterSystem G S]` (orange), `{s : S}` (purple), `invmap S s = s` (red), `:= by` (orange). - **Proof Steps**: 1. `simp [CoxeterSystem.Presentation.invmap]` - `simp` (red), `CoxeterSystem.Presentation.invmap` (green). 2. `unfold CoxeterSystem.toMatrix` - `unfold` (red), `CoxeterSystem.toMatrix` (green). 3. `apply CoxeterSystem.monoidLift.mapLift.of` - `apply` (red), `CoxeterSystem.monoidLift.mapLift.of` (green). ### Detailed Analysis - **Syntax Highlighting**: - Keywords (`lemma`, `simp`, `unfold`, `apply`) are in **red**. - Identifiers (`invmap`, `CoxeterSystem`, `Presentation`, `toMatrix`, `monoidLift`, `mapLift`) are in **green**. - Type annotations (`S:Set G`, `s : S`) are in **blue** and **purple**. - Brackets (`[ ]`, `{ }`) and operators (`:=`, `=`) are in **orange**. ### Key Observations 1. The lemma asserts that the inverse map (`invmap`) of a Coxeter system applied to an element `s` equals `s` itself. 2. The proof uses three steps: - `simp` with `CoxeterSystem.Presentation.invmap` to simplify the expression. - `unfold` to expand the definition of `CoxeterSystem.toMatrix`. - `apply` to invoke `CoxeterSystem.monoidLift.mapLift.of` as the final step. 3. The code relies on domain-specific terms like `CoxeterSystem`, `monoidLift`, and `mapLift`, suggesting a focus on algebraic structures. ### Interpretation This code snippet demonstrates a formal proof in a mathematical or algebraic context, likely verifying a property of Coxeter systems. The use of `simp`, `unfold`, and `apply` indicates a structured approach to theorem proving, common in proof assistants. The lemma’s conclusion (`invmap S s = s`) implies that the inverse map preserves elements in the Coxeter system, a non-trivial result in group theory or combinatorics. The absence of numerical data or visualizations emphasizes the abstract, logical nature of the code. ### Notable Patterns - The proof steps are concise, leveraging built-in tactics (`simp`, `unfold`, `apply`) to automate reasoning. - The code’s reliance on `CoxeterSystem` and `monoidLift` suggests it is part of a larger library for algebraic structures. - The use of `:= by` indicates the proof is deferred to a separate block (not visible in the image). ### Conclusion The image captures a formal proof snippet in a functional programming or proof-assistant environment, focusing on properties of Coxeter systems. The syntax highlighting and structured proof steps highlight the interplay between code and mathematical logic.