Related papers: Greatest HITs: Higher inductive types in coinducti…
In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming…
We present a new model of Guarded Dependent Type Theory (GDTT), a type theory with guarded recursion and multiple clocks in which one can program with, and reason about coinductive types. Productivity of recursively defined coinductive…
Clocked Cubical Type Theory is a new type theory combining the power of guarded recursion with univalence and higher inductive types (HITs). This type theory can be used as a metalanguage for synthetic guarded domain theory in which one can…
We present guarded dependent type theory, gDTT, an extensional dependent type theory with a `later' modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to…
Type theories with multi-clocked guarded recursion provide a flexible framework for programming with coinductive types encoding productivity in types. Combining this with solutions to general guarded domain equations one can also construct…
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract…
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive…
We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive…
Nakano's later modality can be used to specify and define recursive functions which are causal or synchronous; in concert with a notion of clock variable, it is possible to also capture the broader class of productive (co)programs. Until…
Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract…
Nakano's later modality allows types to express that the output of a function does not immediately depend on its input, and thus that computing its fixpoint is safe. This idea, guarded recursion, has proved useful in various contexts, from…
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of…
This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of…
Constructive type theory combines logic and programming in one language. This is useful both for reasoning about programs written in type theory, as well as for reasoning about other programming languages inside type theory. It is…
We consider type inference for guarded recursive data types (GRDTs) -- a recent generalization of algebraic data types. We reduce type inference for GRDTs to unification under a mixed prefix. Thus, we obtain efficient type inference.…
Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by…
Several different topoi have played an important role in the development and applications of synthetic guarded domain theory (SGDT), a new kind of synthetic domain theory that abstracts the concept of guarded recursion frequently employed…
Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the…
A theory of recursive and corecursive definitions has been developed in higher-order logic (HOL) and mechanized using Isabelle. Least fixedpoints express inductive data types such as strict lists; greatest fixedpoints express coinductive…
Coinduction is a widely used technique for establishing behavioural equivalence of programs in higher-order languages. In recent years, the rise of languages with quantitative (e.g.~probabilistic) features has led to extensions of…