Flexible Proof Assistant Tool Enhanced with Sympy for Automated Mathematical Verification

In a follow-up to his earlier work, a researcher has significantly enhanced a proof assistant tool initially designed to automatically verify estimates. The tool was initially a simple proof of concept but has since been transformed into a more sophisticated system capable of handling both propositional logic and a wide range of mathematical tasks. The latest iteration of the proof assistant draws inspiration from the Lean proof assistant and leverages the Python package SymPy for symbolic algebra. How the Proof Assistant Works The tool operates within Python's interactive mode, allowing users to input commands one at a time. This setup is reminiscent of text-based adventure games, providing a straightforward and interactive experience. To start using the proof assistant, one needs to download the relevant files and import them into a Python session: python from main import * Once imported, the tool can be initialized for specific exercises. For instance, to begin a proof involving linear arithmetic, one might use: python p = linarith_exercise() This command presents the user with the initial proof state, including the variables and hypotheses involved, along with the goal to be proven. Example 1: Linear Arithmetic Proof Consider a proof problem where one must demonstrate that if ( x ) and ( y ) are positive real numbers such that ( x < 2y ) and ( y < 3z + 1 ), then ( x < 7z + 2 ). The proof assistant can solve this using the Linarith() tactic: python p.use(Linarith()) For a more detailed explanation, the verbose flag can be set to True: python p.use(Linarith(verbose=True)) This outputs the step-by-step process by which the proof assistant combines the inequalities to achieve the desired result. Example 2: Case Splitting Proof In another scenario, the proof involves case splitting. Suppose the task is to show that the hypotheses ( x > -1 ) and ( x < 1 ), and ( y > -2 ) and ( y < 2 ), imply ( x + y > -3 ) and ( x + y < 3 ). The proof assistant can handle this by breaking down the hypotheses and goals: python p.use(SplitHyp("h1")) p.use(SplitHyp("h2")) p.use(SplitGoal()) p.use(Linarith()) p.use(Linarith()) Each step simplifies part of the problem until the final proof is complete. The tool provides a "pseudo-Lean" description of the proof, outlining the tactics used. Asymptotic Estimation The proof assistant also supports asymptotic estimation. By implementing order of magnitude formalism within SymPy, it can verify asymptotic inequalities. For example, given a positive integer ( N ) and positive real numbers ( x ) and ( y ) such that ( x \leq 2N^2 ) and ( y < 3N ), the goal is to prove ( \Theta(x) \cdot \Theta(y) \leq \Theta(N)^4 ): python p = loglinarith_exercise() p.use(LogLinarith(verbose=True)) The LogLinarith() tactic checks the feasibility of the inequalities and confirms the proof. However, the tool faces challenges when dealing with expressions involving Max functions, as they are treated independently rather than considering their nonlinear relationships. Future Enhancements To address these limitations, the researcher plans to improve the LogLinarith() tactic to handle Max expressions more effectively. Additionally, there are intentions to expand the tool's capabilities to estimate function space norms, incorporating tactics based on lemmas like Hölder's inequality and the Sobolev embedding inequality. SymPy's flexibility allows for the creation of new object classes to facilitate this expansion. Open for Contributions The researcher is confident in the basic framework and is open to suggestions and contributions from the community. New features such as data types, lemmas, and tactics, or example problems that highlight areas for improvement, are welcomed. This collaborative approach aims to make the proof assistant more versatile and powerful, capable of solving a broader range of mathematical tasks. Industry Insights and Company Profiles Industry experts view this development as a significant step forward in automating mathematical proofs, particularly in areas like asymptotic analysis and norm estimation. The integration of advanced symbolic algebra and a user-friendly interactive interface has the potential to streamline research and educational processes. SymPy, a mature and widely-used Python library for symbolic mathematics, has been crucial in enabling this flexibility and efficiency. The proof assistant's similarity to Lean, a well-regarded and powerful proof assistant, suggests that it could serve as a valuable educational tool and research aid in the coming years. Contributions from the community are expected to further refine and expand its functionalities, making it an indispensable tool in various mathematical domains.