Related papers: The Marriage of Univalence and Parametricity
Reynold's parametricity theory captures the property that parametrically polymorphic functions behave uniformly: they produce related results on related instantiations. In dependently-typed programming languages, such relations and…
Libraries of formalized mathematics use a possibly broad range of different representations for a same mathematical concept. Yet light to major manual input from users remains most often required for obtaining the corresponding variants of…
Parametricity is a key metatheoretic property of type systems, which implies strong uniformity & modularity properties of the structure of types within systems possessing it. In recent years, various systems of dependent type theory have…
Multiple types can represent the same concept. For example, lists and trees can both represent sets. Unfortunately, this easily leads to incomplete libraries: some set-operations may only be available on lists, others only on trees.…
In mathematics, it is common practice to have several constructions for the same objects. Mathematicians will identify them modulo isomorphism and will not worry later on which construction they use, as theorems proved for one construction…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We define a computational type theory combining the contentful equality structure of cartesian cubical type theory with internal parametricity primitives. The combined theory supports both univalence and its relational equivalent, which we…
Matching logic is a logical framework for specifying and reasoning about programs using pattern matching semantics. A pattern is made up of a number of structural components and constraints. Structural components are syntactically matched,…
Matching Logic is a framework for specifying programming language semantics and reasoning about programs. Its formulas are called patterns and are built with variables, symbols, connectives and quantifiers. A pattern is a combination of…
Proof assistants play a dual role as programming languages and logical systems. As programming languages, proof assistants offer standard modularity mechanisms such as first-class functions, type polymorphism and modules. As logical…
We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not…
Matching logic is a formalism for specifying, and reasoning about, mathematical structures, using patterns and pattern matching. Growing in popularity, it has been used to define many logical systems such as separation logic with recursive…
We present a refinement of the Calculus of Inductive Constructions in which one can easily define a notion of relational parametricity. It provides a new way to automate proofs in an interactive theorem prover like Coq.
The expression problem describes a fundamental tradeoff between two types of extensibility: extending a type with new operations, such as by pattern matching on an algebraic data type in functional programming, and extending a type with new…
According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible…
We present a unifying framework for type systems for process calculi. The core of the system provides an accurate correspondence between essentially functional processes and linear logic proofs; fragments of this system correspond to…
We present a set of tools for rewriting modulo associativity and commutativity (AC) in Coq, solving a long-standing practical problem. We use two building blocks: first, an extensible reflexive decision procedure for equality modulo AC;…
We exhibit a computational type theory which combines the higher-dimensional structure of cartesian cubical type theory with the internal parametricity primitives of parametric type theory, drawing out the similarities and distinctions…
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and…
Optimizing the design of complex systems requires navigating interdependent decisions, heterogeneous components, and multiple objectives. Our monotone theory of co-design offers a compositional framework for addressing this challenge,…