Related papers: Large and Infinitary Quotient Inductive-Inductive …
This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of…
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…
This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly…
Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they…
We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated…
In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs,…
To ensure decidability and consistency of its type theory, a proof assistant should only accept terminating recursive functions and productive corecursive functions. Most proof assistants enforce this through syntactic conditions, which can…
This is the fourth in a series of papers extending Martin-L\"of's meaning explanation of dependent type theory to higher-dimensional types. In this installment, we show how to define cubical type systems supporting a general schema of…
Generalised algebraic theories (GATs) allow multiple sorts indexed over each other. For example, the theories of categories or Martin-L{\"o}f type theories form GATs. Categories have two sorts, objects and morphisms, and the latter are…
We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two size annotations allowing…
Inductive datatypes in programming languages allow users to define useful data structures such as natural numbers, lists, trees, and others. In this paper we show how inductive datatypes may be added to the quantum programming language QPL.…
Domain theory has been developed as a mathematical theory of computation and to give a denotational semantics to programming languages. It helps us to fix the meaning of language concepts, to understand how programs behave and to reason…
Homotopy type theory is an interpretation of Martin-L\"of's constructive type theory into abstract homotopy theory. There results a link between constructive mathematics and algebraic topology, providing topological semantics for…
In this paper, we define indexed type theories which are related to indexed ($\infty$-)categories in the same way as (homotopy) type theories are related to ($\infty$-)categories. We define several standard constructions for such theories…
Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly…
In sequential functional languages, sized types enable termination checking of programs with complex patterns of recursion in the presence of mixed inductive-coinductive types. In this paper, we adapt sized types and their metatheory to the…
Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of…
In order to avoid well-know paradoxes associated with self-referential definitions, higher-order dependent type theories stratify the theory using a countably infinite hierarchy of universes (also known as sorts), Type$_0$ : Type$_1$ :…
We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a…
The set of integer number lists with finite length, and the set of binary trees with integer labels are both countably infinite. Many inductively defined types also have countably many elements. In this paper, we formalize the syntax of…