2020
DOI: 10.1007/978-3-030-45231-5_14
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Constructing Infinitary Quotient-Inductive Types

Abstract: 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 inductive-inductive definitions, Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definition… Show more

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Cited by 8 publications
(15 citation statements)
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“…We now define our class of higher inductive types that we will construct in Set$\mathsf {Set}$. It will be clear by the definition that this is a special case of QW$QW$‐types [8]. Definition A polynomial$\emph {polynomial}$ is a set A together with a family of sets (Ba)aA$(B_a)_{a \in A}$.…”
Section: Image Preserving Qw$qw$‐typesmentioning
confidence: 99%
See 3 more Smart Citations
“…We now define our class of higher inductive types that we will construct in Set$\mathsf {Set}$. It will be clear by the definition that this is a special case of QW$QW$‐types [8]. Definition A polynomial$\emph {polynomial}$ is a set A together with a family of sets (Ba)aA$(B_a)_{a \in A}$.…”
Section: Image Preserving Qw$qw$‐typesmentioning
confidence: 99%
“…In fact our examples of interest fall within a smaller class with a simpler definition and clearer semantics. This class was studied by Blass [5] under the name free algebras subject to identities and by Fiore, Pitts and Steenkamp in [8] under the name QW$QW$ ‐types (we will refer to them by the latter name). A well known example of such a type is that of “unordered countably branching trees.” We modify the definition of T above to get the higher inductive type TprefixSym$T_{\operatorname{Sym}}$ by adding equations as follows, where we write prefixSymfalse(ωfalse)$\operatorname{Sym}(\omega )$ for the type of permutations ωω$\omega \rightarrow \omega$.…”
Section: Introductionmentioning
confidence: 99%
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“…While Cubical Agda [39] accepts all three HIT definitions, the status of defining HITs in HoTT and Cubical type theories in general is an open research question. Various schemas for defining HITs have been proposed [35,14,11,22,23]. HITs have also been used in other computer science applications [5,24,6,4].…”
Section: Multiset Equalitymentioning
confidence: 99%