\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.