Worked on enhancing the correctness and maintainability of context-free language intersection proofs in the nipkow/AIST repository. Focused on formal verification using Isabelle/HOL and Isabelle/ML, the work involved refactoring and correcting proofs within CFL_intersection.thy to improve the reliability of language membership checks for context-free languages and their intersections. Introduced the derives_start1 lemma, which strengthened the formal guarantees of the existing proof framework. By addressing a key bug and restructuring the proof logic, the changes supported future extensions and ensured that the verification process for context-free language operations remained robust, maintainable, and aligned with best practices in theorem proving.