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Cubical Type Theory Cubical Type Theory is an extension of traditional type theory that introduces new features to better handle higher-dimensional structures and reasoning. In Cubical Type Theory, the key innovation is the use of cubical sets, which provide a geometric interpretation of types and allow for the direct manipulation of paths and higher-dimensional structures. Key features of Cu..

Univalent Foundations Univalent Foundations is a framework for the formalization of mathematics based on homotopy type theory. In Univalent Foundations, the univalence axiom is a key principle that states that isomorphic structures are equal. This principle allows for a more flexible and intuitive treatment of mathematical objects, where equivalences between structures are considered on equal..

Homotopy Type Theory (HoTT) Homotopy Type Theory (HoTT) is a branch of mathematics and a formal system in type theory that establishes connections between types and homotopy theory. In HoTT, types are seen as spaces that can be studied using the techniques of homotopy theory. Key concepts in HoTT include: Identity types: Types that represent equality or paths between elements of a given ..

Type-based Program Analysis and Optimization Type-based program analysis is a technique used to analyze and optimize programs based on their types. By examining the types of expressions and variables in a program, we can derive information about their behavior and usage. Key techniques in type-based program analysis include: Type inference: Deriving types for unannotated expressions and ..

Typed Lambda Calculus and its Applications in Programming Language Theory Typed lambda calculus is an extension of the lambda calculus with a type system, which assigns types to lambda terms. Types provide a way to classify and reason about expressions in lambda calculus, adding a level of safety and structure to the language. In typed lambda calculus, every lambda term has a type, and types ..

Formal Verification and Proof Assistants Formal verification is the process of proving the correctness of software or hardware systems using mathematical methods. It involves specifying the desired properties of a system and then proving that these properties hold under all possible circumstances. Proof assistants are software tools designed to assist in the development and verification of ma..