A Formalisation of a Dependently Typed Language as an Inductive-Recursive Family

Danielsson, Nils Anders

doi:10.1007/978-3-540-74464-1_7

Cited by 30 publications

(38 citation statements)

References 14 publications

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“…The work by Chapman [12] and also the earlier work by Danielsson [13] have a motivation very similar to ours. [13] uses implicit substitutions and this seems rather difficult to use in general, his definitions have rather adhoc character.…”

Section: Related Worksupporting

confidence: 67%

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Type theory in type theory using quotient inductive types

Altenkirch

Kaposi

2016

Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. AbstractWe present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition. We use the elimination principle to define the set-theoretic and logical predicate interpretation. The work has been formalized using the Agda system extended with QITs using postulates.

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Section: Related Worksupporting

confidence: 67%

“…[13] uses implicit substitutions and this seems rather difficult to use in general, his definitions have rather adhoc character. [12] is more principled but relies on setoids and has to add a lot of boilerplate code to characterize families of setoids over a setoid explicitly.…”

Section: Related Workmentioning

confidence: 99%

Type theory in type theory using quotient inductive types

Altenkirch

Kaposi

2016

Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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“…One might think that the category we are looking for is DialgpArg, V q, where Thus, we see that ∆ ε ∆ and ∆ pΓ σq p∆ Γ q wk Γ pσ, ∆q as required. 4 In the same way, the equations for the weakening operation hold.…”

Section: A Category For Inductive-inductive Definitionsmentioning

confidence: 94%

“…Danielsson [4] and Chapman [3] define the syntax of dependent type theory in the theory itself by inductively defining contexts, types in a given context and terms of a given type. Let us concentrate on contexts and types for simplicity.…”

Section: Examples Of Inductive-inductive Definitionsmentioning

confidence: 99%

A Categorical Semantics for Inductive-Inductive Definitions

Altenkirch

Morris

Forsberg

et al. 2011

Algebra and Coalgebra in Computer Science

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Abstract. Induction-induction is a principle for defining data types in Martin-Löf Type Theory. An inductive-inductive definition consists of a set A, together with an A-indexed family B : A Ñ Set, where both A and B are inductively defined in such a way that the constructors for A can refer to B and vice versa. In addition, the constructors for B can refer to the constructors for A. We extend the usual initial algebra semantics for ordinary inductive data types to the inductive-inductive setting by considering dialgebras instead of ordinary algebras. This gives a new and compact formalisation of inductive-inductive definitions, which we prove is equivalent to the usual formulation with elimination rules.

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“…In particular, the model of closures in weak head normal form we will use in Section 3. Recent meta theoretic studies involving explicit substitutions include [Dan07,Cha09,Gra09,ACP09]. In the presence of explicit substitutions, variables are most naturally represented as de Bruijn indices [ACCL91].…”

Section: Introductionmentioning

confidence: 99%

Towards Normalization by Evaluation for the βη-Calculus of Constructions

Abel

2010

Functional and Logic Programming

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Abstract. We consider the Calculus of Constructions with typed beta-eta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as partial equivalence relations between values and type constructors as operators on PERs. This models also yields consistency of the beta-eta-Calculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq.

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A Formalisation of a Dependently Typed Language as an Inductive-Recursive Family

Cited by 30 publications

References 14 publications

Type theory in type theory using quotient inductive types

Type theory in type theory using quotient inductive types

A Categorical Semantics for Inductive-Inductive Definitions

Towards Normalization by Evaluation for the βη-Calculus of Constructions

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