## Diagram: LeanCopilot Tactics Suggestion ### Overview The image shows a screenshot of the LeanCopilot environment, demonstrating the tactic suggestion feature. It highlights how the system can suggest tactics to either finish a proof directly or transform the goals, potentially without completing the proof. ### Components/Axes * **Code Editor (Left):** Contains Lean code with a theorem definition and a "suggest\_tactics" prompt. * `import LeanCopilot` * `theorem add_abc (a b c : Nat) : a + b + c = a + c + b := by` * `suggest_tactics` (highlighted with a blue wavy underline) * **Tactic State (Top-Right):** Indicates the current state of the proof. * `Tactic state` * `No goals` * **Suggestions (Right):** Lists suggested tactics. * `Suggestions` * `Try these:` * `apply Nat.add_right_comm` (Green arrow pointing to this line, with the text "Finish the Proof Directly!") * `rw [Nat.add_assoc ]` * `Remaining subgoals:` * `⊢ a + (b + c) = a + c + b` * `rw [Nat.add_comm ]` * `Remaining subgoals:` * `⊢ c + (a + b) = a + c + b` * `simp [Nat.add_assoc ]` * `Remaining subgoals:` * `⊢ a + (b + c) = a + (c + b)` * **Annotations:** * Orange arrow pointing from the "suggest\_tactics" prompt to the text "Call an LLM to Generate Tactics". * Blue bracket encompassing the last three suggestions, with the text "Transform the Goals Errorlessly But Cannot Finish the Proof". ### Detailed Analysis or ### Content Details The code editor shows the beginning of a Lean theorem definition named `add_abc`. The theorem states that for natural numbers `a`, `b`, and `c`, `a + b + c = a + c + b`. The `suggest_tactics` prompt indicates that the user is requesting tactic suggestions from LeanCopilot. The tactic state shows "No goals", indicating that the proof is either complete or that the current state has no remaining subgoals. The suggestions section provides a list of tactics that LeanCopilot recommends. The first suggestion, `apply Nat.add_right_comm`, is highlighted as a tactic that can directly finish the proof. The subsequent suggestions involve rewriting and simplification using associativity and commutativity properties of natural number addition. These tactics transform the goals but do not necessarily complete the proof. ### Key Observations * LeanCopilot suggests tactics to both directly finish the proof and transform the goals. * The "apply" tactic is presented as a direct solution, while "rw" and "simp" tactics are presented as transformations. * The annotations clarify the purpose of the "suggest\_tactics" prompt and the effect of the suggested tactics. ### Interpretation The image demonstrates the functionality of LeanCopilot in suggesting tactics for Lean theorem proving. It shows that the system can provide tactics that either directly solve the problem or transform it into a different, potentially simpler, form. The annotations highlight the role of the LLM in generating tactics and the potential for tactics to transform goals without necessarily finishing the proof. The image suggests that LeanCopilot can be a valuable tool for Lean users by providing intelligent tactic suggestions.