Indexed containers

Altenkirch, Thorsten; Ghani, Neil; Hancock, Peter; McBride, Conor; Morris, Peter D.

doi:10.1017/s095679681500009x

Cited by 48 publications

(50 citation statements)

References 28 publications

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“…(∃n.F A n z) → F (λz ′ .∃n.A n z ′ ) z is an isomorphism for every A. All finitely branching indexed containers [Altenkirch et al 2006], i.e. functors of the form F X z ≡ Σ(c : C z).…”

Section: Isomorphisms For Sizementioning

confidence: 99%

Parametric quantifiers for dependent type theory

Nuyts

Vezzosi

Devriese

2017

Proc. ACM Program. Lang.

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Polymorphic type systems such as System F enjoy the parametricity property: polymorphic functions cannot inspect their type argument and will therefore apply the same algorithm to any type they are instantiated on. This idea is formalized mathematically in Reynolds's theory of relational parametricity, which allows the metatheoretical derivation of parametricity theorems about all values of a given type. Although predicative System F embeds into dependent type systems such as Martin-Löf Type Theory (MLTT), parametricity does not carry over as easily. The identity extension lemma, which is crucial if we want to prove theorems involving equality, has only been shown to hold for small types, excluding the universe. We attribute this to the fact that MLTT uses a single type former Π to generalize both the parametric quantifier ∀ and the type former → which is non-parametric in the sense that its elements may use their argument as a value. We equip MLTT with parametric quantifiers ∀ and ∃ alongside the existing Π and Σ, and provide relation type formers for proving parametricity theorems internally. We show internally the existence of initial algebras and final co-algebras of indexed functors both by Church encoding and, for a large class of functors, by using sized types. We prove soundness of our type system by enhancing existing iterated reflexive graph (cubical set) models of dependently typed parametricity by distinguishing between edges that express relatedness of objects (bridges) and edges that express equality (paths). The parametric functions are those that map bridges to paths. We implement an extension to the Agda proof assistant that type-checks proofs in our type system.

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Section: Isomorphisms For Sizementioning

confidence: 99%

Parametric quantifiers for dependent type theory

Nuyts

Vezzosi

Devriese

2017

Proc. ACM Program. Lang.

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“…In this section, we make use of indexed higher inductive types, and this is not part of our formalization. Note that indexed inductive types can always be encoded via inductive types [2,33], and we expect that the same is true for indexed higher inductive types. 5.1.…”

Section: Free Groupoids and A Higher Seifert-van Kampen Theoremmentioning

confidence: 98%

Path Spaces of Higher Inductive Types in Homotopy Type Theory

Kraus

Raumer

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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The study of equality types is central to homotopy type theory. Characterizing these types is often tricky, and various strategies, such as the encode-decode method, have been developed.We prove a theorem about equality types of coequalizers and pushouts, reminiscent of an induction principle and without any restrictions on the truncation levels. This result makes it possible to reason directly about certain equality types and to streamline existing proofs by eliminating the necessity of auxiliary constructions.To demonstrate this, we give a very short argument for the calculation of the fundamental group of the circle (Licata and Shulman [26]), and for the fact that pushouts preserve embeddings. Further, our development suggests a higher version of the Seifert-van Kampen theorem, and the set-truncation operator maps it to the standard Seifert-van Kampen theorem (due to Favonia and Shulman [18]).

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“…Generalizations of the early presheaf representations have been developed using the (related) concepts of dependent polynomial functors [Gambino and Hyland 2003], generalized species [Fiore et al 2008], polynomial functors over grupoids [Kock 2012], and indexed containers [Altenkirch et al 2015;Morris et al 2009]. Indexed containers feature a stronger type-theoretic view that complements the category-theoretic view.…”

Section: Category-theoretic Approachesmentioning

confidence: 99%

Bindings as bounded natural functors

Blanchette

Gheri

Popescu

et al. 2019

Proc. ACM Program. Lang.

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We present a general framework for specifying and reasoning about syntax with bindings. Abstract binder types are modeled using a universe of functors on sets, subject to a number of operations that can be used to construct complex binding patterns and binding-aware datatypes, including non-well-founded and infinitely branching types, in a modular fashion. Despite not committing to any syntactic format, the framework is łconcretež enough to provide definitions of the fundamental operators on terms (free variables, alpha-equivalence, and capture-avoiding substitution) and reasoning and definition principles. This work is compatible with classical higher-order logic and has been formalized in the proof assistant Isabelle/HOL.

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Indexed containers

Cited by 48 publications

References 28 publications

Parametric quantifiers for dependent type theory

Parametric quantifiers for dependent type theory

Path Spaces of Higher Inductive Types in Homotopy Type Theory

Bindings as bounded natural functors

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