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Explore a 20-minute conference talk from POPL 2025 that introduces MimIR, an extensible and type-safe intermediate representation designed for domain-specific languages (DSLs). Learn how traditional compilers struggle with modern computation-heavy applications in fields like image processing, machine learning, and numerical simulations, and how MimIR addresses these limitations. The presentation explains how MimIR functions as a pure type system based on typed lambda calculus, allowing developers to declare new domain-specific operations called 'axioms' and extend the compiler through plugins. The speakers demonstrate MimIR's effectiveness through three case studies: low-level plugins operating at LLVM's abstraction level, a regular-expression matching plugin, and plugins for linear algebra and automatic differentiation—all producing state-of-the-art performance. Presented by researchers from the University of Mannheim and Saarland University at the ACM SIGPLAN conference.