## Screenshot: LeanCopilot Tactic Suggestion ### Overview The image is a screenshot of the LeanCopilot interface, showcasing a tactic suggestion for proving a theorem. It highlights the use of LLM-based tactic suggestions paired with best-first search to find a complete proof. ### Components/Axes * **Code Editor**: Displays the Lean code, including the theorem definition and the "search\_proof" command. * **Tactic State**: Shows the current state of the proof, indicating "No goals". * **Suggestions**: Provides tactic suggestions to the user. * **Annotations**: Arrows and text boxes highlight specific elements and provide explanations. ### Detailed Analysis or ### Content Details 1. **Code Snippet**: * `import LeanCopilot` * `theorem add_abc (a b c : Nat) : a + b + c = c + b + a := by` * `search_proof` (highlighted with an orange box and wavy underline) 2. **Tactic State**: * `Tactic state` (with a downward-pointing triangle) * `No goals` 3. **Suggestions**: * `Suggestions` (with a downward-pointing triangle) * `Try this:` * `simp [Nat.add_comm, Nat.add_left_comm]` (enclosed in a rounded blue box) 4. **Annotations**: * An orange arrow points from the "search\_proof" line to the text: "Pair LLM-Based Tactic Suggestion with Best-First Search". * A blue arrow points from the right side of the code to the text: "Complete Proof Found". ### Key Observations * The LeanCopilot suggests the `simp` tactic with `Nat.add_comm` and `Nat.add_left_comm` to prove the theorem. * The system indicates that a complete proof has been found. * The "search\_proof" command triggers the tactic suggestion mechanism. ### Interpretation The screenshot demonstrates the LeanCopilot's ability to suggest relevant tactics for proving theorems in Lean. The combination of LLM-based suggestions and best-first search appears to be effective in finding complete proofs. The "No goals" state indicates that the suggested tactic successfully closed the proof. The use of `Nat.add_comm` (commutativity of addition) and `Nat.add_left_comm` (left commutativity of addition) suggests that the proof relies on rearranging the terms in the addition expression.

\n ## Screenshot: LeanCopilot Interface - Theorem Proof Suggestion ### Overview This screenshot depicts the LeanCopilot interface, showcasing a theorem proof suggestion within a formal proof environment. The interface highlights a theorem, a search for a proof, and a suggested tactic to complete the proof. The right-hand side displays the tactic state and suggestions. ### Components/Axes The screenshot contains the following elements: * **Code Block:** Displays the theorem definition. * **Annotation Arrows:** Indicate the flow and purpose of different elements. * **Tactic State Panel:** Shows the current state of the proof tactic. * **Suggestions Panel:** Provides suggested tactics for proof completion. ### Content Details The code block contains the following: * `import LeanCopilot` * `theorem add_abc (a b c : Nat) : a + b + c = c + b + a := by` * `search proof` The Tactic State Panel displays: * **Tactic state:** "No goals" * **Suggestions:** "Try this:" The Suggestions Panel displays: * `simp [Nat.add_comm, Nat.add_left_comm]` Annotation Arrows: * An arrow points from `search proof` to the right side of the screen, labeled "Pair LLM-Based Tactic Suggestion with Best-First Search". * An arrow points to the right side of the screen, labeled "Complete Proof Found". ### Key Observations The screenshot demonstrates a successful proof suggestion by LeanCopilot. The "No goals" message in the Tactic State Panel indicates that the theorem has been successfully proven using the suggested tactic. The annotation arrows emphasize the role of the LLM-based tactic suggestion in finding the proof. ### Interpretation The screenshot illustrates the functionality of LeanCopilot as an automated theorem prover. The system takes a theorem definition and a search for a proof as input, and then suggests a tactic to complete the proof. The successful completion of the proof, as indicated by the "No goals" message, demonstrates the effectiveness of the system. The use of `simp` (simplification) with `Nat.add_comm` (natural number addition commutativity) and `Nat.add_left_comm` (natural number addition left commutativity) suggests that the proof relies on the commutative property of addition. The LLM-based tactic suggestion and best-first search indicate that the system employs advanced search algorithms to find the optimal proof strategy. This suggests a powerful tool for formal verification and mathematical reasoning.

\n ## Screenshot: Lean 4 Theorem Proving with LeanCopilot AI Assistant ### Overview This image is a screenshot of a code editor or Integrated Development Environment (IDE) interface, specifically demonstrating the use of a tool called "LeanCopilot" within the Lean 4 theorem prover. The screenshot is annotated to highlight how the tool pairs a Large Language Model (LLM) with a search algorithm to automatically suggest and complete formal mathematical proofs. ### Components/Axes The image is segmented into three primary regions: 1. **Main Code Editor (Left/Center):** Contains Lean 4 source code. 2. **Annotation Overlays:** Orange and blue text and arrows added to explain functionality. 3. **IDE Sidebar (Right):** A panel showing the current proof state and AI-generated suggestions. **Textual Content & Labels:** * **Code Editor Content:** * Line 1: `import LeanCopilot` * Line 3: `theorem add_abc (a b c : Nat) : a + b + c = c + b + a := by` * Line 4 (within a highlighted box): `search_proof` * **Annotation Overlays:** * **Orange Annotation:** An orange box surrounds the text `search_proof`. An orange arrow points from this box to the orange text below: `Pair LLM-Based Tactic Suggestion with Best-First Search`. * **Blue Annotation:** Blue text states `Complete Proof Found`. A blue arrow points from this text to a blue-outlined box in the sidebar. * **IDE Sidebar (Right Panel):** * **Header 1:** `▼ Tactic state` * Sub-item: `No goals` (displayed in blue text). * **Header 2:** `▼ Suggestions` * Sub-header: `Try this:` * **Suggestion Box (Blue Outline):** Contains the tactic: ``` simp [Nat.add_comm, Nat.add_left_comm] ``` ### Detailed Analysis The screenshot captures a specific moment in a formal proof development workflow. 1. **Theorem Statement:** The user has defined a theorem named `add_abc` which asserts the commutativity of addition for three natural numbers (`Nat`): `a + b + c = c + b + a`. The proof begins with the `by` keyword. 2. **AI Integration Point:** The tactic `search_proof` is entered. This is not a standard Lean tactic but is provided by the imported `LeanCopilot` library. The annotation clarifies that this command initiates a process that combines LLM-based tactic suggestions with a best-first search algorithm. 3. **Proof State:** The sidebar's "Tactic state" section shows `No goals`. This indicates that after the `search_proof` command was executed, the AI system successfully found a sequence of tactics that closed all remaining proof goals, completing the theorem. 4. **AI Suggestion:** The "Suggestions" panel shows the specific tactic the AI system identified to complete the proof: `simp [Nat.add_comm, Nat.add_left_comm]`. This is a simplification (`simp`) tactic using two specific lemmas about the commutativity of natural number addition. 5. **Visual Workflow:** The annotations create a clear visual flow: The user invokes `search_proof` (orange), which triggers the AI system. The system then finds and applies a complete proof (blue), with the specific tactic suggestion displayed in the sidebar. ### Key Observations * **Language:** The primary language in the code is **Lean 4**, a functional programming language and theorem prover. * **Toolchain:** The image showcases the `LeanCopilot` library, which acts as a bridge between the Lean prover and external AI models. * **Automation Level:** The system demonstrates a high level of automation. The user only needed to state the theorem and invoke `search_proof`; the AI handled the discovery of the proof steps. * **Proof Method:** The suggested proof relies on existing library lemmas (`Nat.add_comm`, `Nat.add_left_comm`) rather than manual induction, indicating the AI can leverage a knowledge base of formal mathematics. * **UI State:** The "No goals" message is the definitive indicator of a successfully completed proof in Lean. ### Interpretation This screenshot is a practical demonstration of **AI-assisted formal verification**. It illustrates a paradigm where a human provides the high-level intent (the theorem statement), and an AI agent handles the low-level, often tedious, logical steps required to construct a formal proof. The key relationship shown is between the user's command (`search_proof`) and the AI's output (the `simp` tactic). The "Complete Proof Found" message signifies that the AI's suggested tactic, when applied, was sufficient to satisfy the theorem's logical requirements completely. This has significant implications for making formal methods more accessible and efficient, as it reduces the expertise and manual effort needed to produce machine-checked proofs. The image serves as both a technical documentation of the `LeanCopilot` tool's interface and a conceptual showcase of human-AI collaboration in mathematical reasoning.

## Screenshot: LeanCopilot Theorem Prover Interface ### Overview The image shows a LeanCopilot interface for formal theorem proving in Lean 4. The screen displays a theorem definition (`add_abc`) with a partially completed proof, annotated with LLM-generated suggestions and a "Complete Proof Found" indicator. Key elements include code syntax highlighting, tactic state tracking, and interactive proof guidance. ### Components/Axes 1. **Code Editor**: - **Text**: - `import LeanCopilot` (blue syntax highlighting) - `theorem add_abc (a b c : Nat) : a + b + c = c + b + a := by` (black text with blue keywords) - **Annotations**: - Orange box around `search_proof` with arrow labeled *"Pair LLM-Based Tactic Suggestion with Best-First Search"* - Blue arrow labeled *"Complete Proof Found"* pointing to a suggestion box 2. **Tactic State Panel**: - **Labels**: - `Tactic state` (black text) - `No goals` (blue text) - `Suggestions` (black text) - **Content**: - Dropdown suggestion: `simp [Nat.add_comm, Nat.add_left_comm]` (blue text) 3. **Visual Indicators**: - Red vertical bars on the left and right margins (likely indicating cursor position or proof progress) - Blue highlight under `add_abc` (theorem name) ### Detailed Analysis - **Theorem Definition**: The theorem `add_abc` asserts the commutativity of addition for three natural numbers. The proof attempt (`:= by`) invokes LeanCopilot’s LLM-powered proof assistant. - **LLM Suggestion**: The `search_proof` function (highlighted in orange) is described as pairing an LLM-based tactic suggestion system with a best-first search algorithm. This implies the system prioritizes high-priority proof steps (e.g., symmetry/associativity laws) to efficiently find a proof. - **Tactic State**: The `Tactic state` panel shows no active subgoals (`No goals`), indicating the theorem is fully proven. The `Suggestions` section recommends using `simp` with commutativity axioms (`Nat.add_comm`, `Nat.add_left_comm`), which the LLM identified as critical for completing the proof. - **Proof Completion**: The blue arrow labeled *"Complete Proof Found"* connects the suggestion box to the theorem, indicating the system successfully automated the proof using the suggested tactics. ### Key Observations 1. **Syntax Highlighting**: - Keywords (`import`, `theorem`, `Nat`, `:= by`) are highlighted in blue, while variable names (`a`, `b`, `c`) are in black. 2. **Interactive Guidance**: - The LLM suggests specific `simp` tactics, which the user can apply directly via the dropdown. 3. **Proof Automation**: - The system’s ability to find a complete proof (`Complete Proof Found`) demonstrates the integration of LLM-generated suggestions with Lean’s proof engine. ### Interpretation This interface illustrates how LeanCopilot combines formal theorem proving with LLM-driven automation. The best-first search strategy likely prioritizes tactics that reduce proof complexity (e.g., commutativity axioms), enabling the system to efficiently navigate the proof space. The absence of subgoals (`No goals`) confirms the theorem’s validity, while the `simp` suggestion highlights the role of algebraic axioms in automated reasoning. The design emphasizes user interaction with LLM-generated hints, bridging the gap between human intuition and formal verification. **Note**: No numerical data or trends are present; the focus is on textual and structural elements of the proof assistant interface.