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Explore an intrinsic representation of type theory in Cubical Agda through this 23-minute conference presentation from CPP 2026. Learn how researchers Liang-Ting Chen, Fredrik Nordvall Forsberg, and Tzu-Chun Tsai developed a formalization inspired by Awodey's natural models of type theory, using quotient inductive-inductive-recursive types to create syntax that Cubical Agda accepts without requiring transports, postulates, or custom rewrite rules. Discover their formalization of key meta-properties including the standard model, normalisation by evaluation for typed terms, and strictification constructions, all implemented using Cubical Agda's native support for quotient inductive types to ensure reasonable computational speed. Understand the challenges encountered when developing more sophisticated metatheory, particularly the reemergence of the 'transport hell' problem, and examine why developing type theory metatheory using intrinsic representations without strict equations remains a considerable struggle regardless of whether natural models are employed. Gain insights into the practical limitations and theoretical implications of this approach to formalizing type theory in proof assistants.
