Type theory in type theory using quotient inductive types

AltenkirchThorsten,; KaposiAmbrus,

doi:10.1145/2914770.2837638

Cited by 10 publications

(14 citation statements)

References 16 publications

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“…the very first sentence by Abel et al [9]), which refers to formalising the (intended) initial model of type theory. 1 Altenkirch and Kaposi [3] call this simply type theory in type theory. To avoid confusion, we refer to the type theory in which these models are implemented as the host theory, and the structure that get implemented as the object theory or simply the model.…”

Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

“…1: The formulation of a category with families (CwF) as a generalised algebraic theory (GAT), as given by Kaposi and others [1], [2]. Presentation-wise, it slightly differs from (but is equivalent to) the similar suggestion by Altenkirch and Kaposi [3]. Above, the components are reordered and regrouped to make the connection to CwF's more visible.…”

Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

“…For example, given a type A : Ty ∆ and a substitution σ : Sub Γ ∆, we need an operation (often written as [ ]) which gives us a new type A[σ] : Ty Γ. Equalities are stated using Martin-Löf's identity type, also known as equality type, path type, or identification type, denoted by a = b. 3 As an example, we need an equality of the form…”

Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

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Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

Kraus

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Using dependent type theory to formalise the syntax of dependent type theory is a very active topic of study and goes under the name of "type theory eating itself" or "type theory in type theory." Most approaches are at least loosely based on Dybjer's categories with families (CwF's) and come with a type Con of contexts, a type family Ty indexed over it modelling types, and so on. This works well in versions of type theory where the principle of unique identity proofs (UIP) holds. In homotopy type theory (HoTT) however, it is a long-standing and frequently discussed open problem whether the type theory "eats itself" and can serve as its own interpreter. The fundamental underlying difficulty seems to be that categories are not suitable to capture a type theory in the absence of UIP.In this paper, we develop a notion of ∞-categories with families (∞-CwF's). The approach to higher categories used relies on the previously suggested semi-Segal types, with a new construction of identity substitutions that allow for both univalent and non-univalent variations. The type-theoretic universe as well as the internalised (set-level) syntax are models, although it remains a conjecture that the latter is initial. To circumvent the known unsolved problem of constructing semisimplicial types, the definition is presented in two-level type theory (2LTT).Apart from introducing ∞-CwF's, the paper explains the shortcomings of 1-categories in type theory without UIP as well as the difficulties of and approaches to internal higher-dimensional categories.

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Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

Section: Introduction: Formalising Type Theorymentioning

confidence: 99%

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Internal ∞-Categorical Models of Dependent Type Theory : Towards 2LTT Eating HoTT

Kraus

2021

2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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“…In the sense of HoTT we mean a type theory limited to h-sets 6. A strict model is one where every equation holds definitionally.…”

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confidence: 99%

“…The main example of induction-induction is the intrinsic definition of a dependent type theory in type theory[6].…”

mentioning

confidence: 99%

Constructing a universe for the setoid model

Altenkirch

Boulier

Kaposi

et al. 2021

Lecture Notes in Computer Science

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The setoid model is a model of intensional type theory that validates certain extensionality principles, like function extensionality and propositional extensionality, the latter being a limited form of univalence that equates logically equivalent propositions. The appeal of this model construction is that it can be constructed in a small, intensional, type theoretic metatheory, therefore giving a method to boostrap extensionality. The setoid model has been recently adapted into a formal system, namely Setoid Type Theory (SeTT). SeTT is an extension of intensional Martin-Löf type theory with constructs that give full access to the extensionality principles that hold in the setoid model.Although already a rich theory as currently defined, SeTT currently lacks a way to internalize the notion of type beyond propositions, hence we want to extend SeTT with a universe of setoids. To this aim, we present the construction of a (non-univalent) universe of setoids within the setoid model, first as an inductive-recursive definition, which is then translated to an inductive-inductive definition and finally to an inductive family. These translations from more powerful definition schemas to simpler ones ensure that our construction can still be defined in a relatively small metatheory which includes a proof-irrelevant identity type with a strong transport rule.

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