Small Induction Recursion

Hancock, Peter; McBride, Conor; Ghani, Neil; Malatesta, L.; Altenkirch, Thorsten

doi:10.1007/978-3-642-38946-7_13

Cited by 14 publications

(9 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Indexed inductive types (which are used in Nordvall Forsberg's construction) are constructed from plain inductive types in Altenkirch et al [3], with good computational properties (provided an identity type that satisfies J strictly). And small induction-recursion is reduced to plain indexed inductive types in Hancock et al [11].…”

Section: Related Workmentioning

confidence: 99%

Constructing Inductive-Inductive Types in Cubical Type Theory

Hugunin

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Inductive-inductive types are a joint generalization of mutual inductive types and indexed inductive types. In extensional type theory, inductive-inductive types can be constructed from inductive types, and this construction has been conjectured to work in intensional type theory as well. In this paper, we show that the existing construction requires Uniqueness of Identity Proofs, and present a new construction (which we conjecture generalizes) of one particular inductive-inductive type in cubical type theory, which is compatible with homotopy type theory.

show abstract

Section: Related Workmentioning

confidence: 99%

Constructing Inductive-Inductive Types in Cubical Type Theory

Hugunin

2019

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…The reduction of indexed inductive-definitions to Petersson-Synek Trees has been developed in container theory, see esp. [19,14] but as well [6,21]. There are various set theoretic models of final coalgebras, examples are de Bruin [8], Barr [7] or Aczel [4].…”

Section: Related Workmentioning

confidence: 99%

“…They are initial algebras of indexed containers in the theory of containers, see [6,21]. In [14,19] a formal proof that initial algebras of indexed containers and therefore Petersson-Synek trees subsume all indexed inductive definitions is given. We write in the following Tree instead of D and tree for the constructor C. Let us fix in the following A, B, j: Assumption 3.1 (a) In the following assume…”

Section: Initial Algebras and Inductively Defined Setsmentioning

confidence: 99%

How to Reason Coinductively Informally

Setzer

2016

Advances in Proof Theory

View full text Add to dashboard Cite

We start by giving an overview of the theory of indexed inductively and coinductively defined sets. We consider the theory of strictly positive indexed inductive definitions in a set theoretic setting. We show the equivalence between the definition as an indexed initial algebra, the definition via an induction principle, and the set theoretic definition of indexed inductive definitions. We review as well the equivalence of unique iteration, unique primitive recursion, and induction. Then we review the theory of indexed coinductively defined sets or final coalgebras. We construct indexed coinductively defined sets set theoretically, and show the equivalence between the category theoretic definition, the principle of unique coiteration, of unique corecursion, and of iteration together with bisimulation as equality. Bisimulation will be defined as an indexed coinductively defined set. Therefore proofs of bisimulation can be carried out corecursively. This fact can be considered together with bisimulation implying equality as the coinduction principle for the underlying coinductively defined set. Finally we introduce various schemata for reasoning about coinductively defined sets in an informal way: the schemata of corecursion, of indexed corecursion, of coinduction, and of corecursion for coinductively defined relations. This allows to reason about coinductively defined sets similarly as one does when reasoning about inductively defined sets using schemata of induction. We obtain the notion of a coinduction hypothesis, which is the dual of an induction hypothesis.

show abstract

“…Since the publication of the LICS paper, indexed containers have been used as a base for the generic definition of datatypes for Epigram 2 (Chapman et al 2010), and to develop the theory of ornaments (McBride 2010). In recent work, it has been shown that indexed containers are sufficient to express all small inductive-recursive definitions (Hancock et al 2013)…”

Section: Related Workmentioning

confidence: 99%

Indexed containers

et al. 2015

Self Cite

View full text Add to dashboard Cite

We show that the syntactically rich notion of strictly positive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. As a result, we show indexed containers provide normal forms for strictly positive families in much the same way that containers provide normal forms for strictly positive types. Interestingly, this step from containers to indexed containers is achieved without having to extend the core type theory. Most of the construction presented here has been formalized using the Agda system.

show abstract

Small Induction Recursion

Cited by 14 publications

References 16 publications

Constructing Inductive-Inductive Types in Cubical Type Theory

Constructing Inductive-Inductive Types in Cubical Type Theory

How to Reason Coinductively Informally

Indexed containers

Contact Info

Product

Resources

About