## Flow Diagram: APOLLO Framework ### Overview The image presents a flow diagram illustrating the APOLLO Framework, likely used for formal mathematical proofs, showcasing the different stages and tools involved in verifying a theorem. The diagram consists of multiple boxes, each representing a step or a tool, connected by arrows that indicate the flow of information. ### Components/Axes * **Title:** APOLLO Framework (centered at the top) * Boxes: Each box represents a module or step in the framework. The boxes are color-coded and arranged in a top-down flow, with arrows indicating the direction of processing. Modules include "LLM," "Syntax Refiner", "Sorrifier", and "Auto Solver". * Arrows: Black arrows indicate the flow of data between the modules. * Text: Each box contains Lean code with theorem statements and proof tactics. * Color Coding (Top-Left of the Box): * Red: LLM (Large Language Model) * Orange: Syntax Refiner * Yellow: Sorrifier * Blue: Auto Solver * Green: LLM + APOLLO, Base Proof + Lemma #5 + Lemma #6 ### Detailed Analysis or ### Content Details **1. Initial Setup (Top of Diagram)** * **LLM (Red Box):** * Imports: `Mathlib`, `Aesop` * Options: * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * Opens: `BigOperators Real Nat Topology Rat` * Theorem Statement (Informal): Real numbers 'x' and 'y' have an arithmetic mean of 7 and a geometric mean of sqrt(19). Show that x^2 + y^2 = 158. * Theorem Declaration: `theorem mathd_algebra_332 (x y : R) (h₀ : (x + y) / 2 = (7 : R)) (h₁ : Real.sqrt (x * y) = Real.sqrt (19 : R)) : x^2 + y^2 = (158 : R) := by` * Proof starts by rewriting the sum as x+y=14 * It substitues the given equation * It proves that x*y > 0 * It uses injectivity of sqrt on nonnegatives to deduce x * y = 19 * Finally, it substitutes and simplifies using ring * **Syntax Refiner (Orange Box):** * Imports: `Mathlib`, `Aesop` * Options: * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * Opens: `BigOperators Real Nat Topology Rat` * Theorem Declaration: `theorem mathd_algebra_332 (x y : R) (h₀ : (x + y) / 2 = (7 : R)) (h₁ : Real.sqrt (x * y) = Real.sqrt (19 : R)) : x^2 + y^2 = (158 : R) := by` * Proof starts by rewriting the sum as x+y=14 * It substitues the given equation * It proves that x*y > 0 * The proof uses `by` and `ring` * The proof also applies `sqrt_inj` * **Sorrifier (Yellow Box):** * Imports: `Mathlib`, `Aesop` * Options: * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * Opens: `BigOperators Real Nat Topology Rat` * Theorem Declaration: `theorem mathd_algebra_332 (x y : R) (h₀ : (x + y) / 2 = (7 : R)) (h₁ : Real.sqrt (x * y) = Real.sqrt (19 : R)) : x^2 + y^2 = (158 : R) := by` * Proof starts by rewriting the sum as x+y=14 * It substitues the given equation * It proves that x*y > 0 * There are `sorry` statements here that need to be refined. * **Auto Solver (Blue Box):** * Imports: `Mathlib`, `Aesop` * Options: * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * Opens: `BigOperators Real Nat Topology Rat` * Theorem Declaration: `theorem mathd_algebra_332 (x y : R) (h₀ : (x + y) / 2 = (7 : R)) (h₁ : Real.sqrt (x * y) = Real.sqrt (19 : R)) : x^2 + y^2 = (158 : R) := by` * Proof uses: * `linarith` * `simp_all only [Real.sqrt_pos, ofNat_pos]` **2. Intermediate Stages (Middle of Diagram)** * Multiple iterations of lemma definitions, refining tactics, and utilizing different modules ("LLM", "Syntax Refiner", "Sorrifier", "Auto Solver") * Lemma Definition 1: `lemma mathd_algebra_332_1 (x y : R) (h₁ : (x * y) = (19 : R)) (sum : x + y = (14 : R)) (prod_pos : (0 : R) < x * y) (h₀ : True) : x * y = (19 : R) := by` **3. Base Proof and Final Result (Bottom of Diagram)** * **Base Proof + Lemma #5 + Lemma #6 (White Box):** Indicates that the base proof relies on Lemma #5 and Lemma #6 * **LLM + APOLLO (Green Box):** * Imports: `Mathlib`, `Aesop` * Options: * `set_option maxHeartbeats 0` * `set_option pp.numericTypes true` * `set_option pp.coercions.types true` * Opens: `BigOperators Real Nat Topology Rat` * Theorem Declaration: `theorem mathd_algebra_332 (x y : R) (h₀ : (x + y) / 2 = (7 : R)) (h₁ : Real.sqrt (x * y) = Real.sqrt (19 : R)) : x^2 + y^2 = (158 : R) := by` **4. Formal Statement (Bottom Left of Diagram)** * A restatement of the problem being solved: Given real numbers x and y with an arithmetic mean of 7 and a geometric mean of sqrt(19), show that x^2 + y^2 = 158. ### Key Observations * The diagram depicts a structured approach to proving a mathematical theorem, leveraging different automated tools and lemmas to arrive at the final result. * The flow starts with an initial statement and progresses through refinement stages, utilizing tools like "Syntax Refiner," "Sorrifier", and "Auto Solver." * The use of color-coded boxes helps to visually distinguish the roles of different components in the framework. * The repeated appearance of the same theorem declaration across different modules suggests an iterative process of refining the proof. * The presence of "sorry" statements in the intermediate steps indicates incomplete or placeholder proofs. ### Interpretation The diagram illustrates a framework for formalizing and verifying mathematical proofs using a combination of AI (LLM) and automated reasoning tools. The process begins with an informal statement, which is then translated into formal Lean code. The proof is refined through multiple stages, leveraging tactics provided by different modules, until a complete and verified proof is achieved. This suggests an approach to theorem proving where AI and automated tools collaborate to handle complex mathematical reasoning. The diagram underlines the importance of different proof tactics and strategies in achieving a conclusive result.