Constructing quotient inductive-inductive types

Authors

Ambrus Kaposi, András Kovács, Thorsten Altenkirch

Type: Article

Publication Date: 2019-01-02

Citations: 37

DOI: https://doi.org/10.1145/3290315

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Abstract

Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting with uniqueness of identity proofs, hence we use the term QIIT instead of higher inductive-inductive type. An example of a QIIT is the well-typed (intrinsic) syntax of type theory quotiented by conversion. In this paper first we specify finitary QIITs using a domain-specific type theory which we call the theory of signatures. The syntax of the theory of signatures is given by a QIIT as well. Then, using this syntax we show that all specified QIITs exist and they have a dependent elimination principle. We also show that algebras of a signature form a category with families (CwF) and use the internal language of this CwF to show that dependent elimination is equivalent to initiality.

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Proceedings of the ACM on Programming Languages
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Repository@Nottingham (University of Nottingham)
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Summary

Large and Infinitary Quotient Inductive-Inductive Types

2020-05-26

András Kovács , Ambrus Kaposi

Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic … Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QIITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QIITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

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Type-Theoretic Signatures for Algebraic Theories and Inductive Types

2022-01-01

András Kovács

Summary of Thesis This thesis develops the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type … Summary of Thesis This thesis develops the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. In the thesis, we describe three type theories where each typing context can be viewed as an algebraic signature, specifying sorts, operations and equations. These signatures are useful in broader mathematical contexts, but we are also concerned with potential implementation in proof assistants. In Chapter 3, we describe a way to use two-level type theory as a metalanguage for developing semantics of algebraic signatures. This makes it possible to work in a concise internal notation of a type theory, and at the same time build semantics internally to arbitrary structured categories. For example, the signature for natural number objects can be interpreted in any category with finite products. In Chapter 4, we describe finitary quotient inductive-inductive (FQII) signatures. Most type theories themselves can be specified with FQII signatures. We build a structured category of algebras for each signature, where equivalence of initiality and induction can be shown. We additionally present term algebra constructions, constructions of left adjoint functors of signature morphisms, and we describe a way to use self-describing signatures to minimize necessary metatheoretic assumptions. In Chapter 5, we describe infinitary quotient inductive-inductive signatures. These allow specification of infinitely branching trees as initial algebras. We adapt the semantics from the previous chapter. We also revisit term models, left adjoints of signature morphisms and self-description of signatures. We also describe how to build semantics of signatures internally to the theory of signatures itself, which yields numerous ways to build new signatures from existing ones. In Chapter 6, we describe higher inductive-inductive signatures. These differ from previous semantics mostly in that their intended semantics is in homotopy type theory, and allows higher-dimensional equalities. In this more general setting we only consider enough semantics to compute notions of initiality and induction for each signature.

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Type-Theoretic Signatures for Algebraic Theories and Inductive Types

2023-01-01

András Kovács

We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in … We develop the usage of certain type theories as specification languages for algebraic theories and inductive types. We observe that the expressive power of dependent type theories proves useful in the specification of more complicated algebraic theories. We describe syntax and semantics for three classes of algebraic theories: finitary quotient inductive-inductive theories, their infinitary generalization, and finally higher inductive-inductive theories. In each case, an algebraic signature is a typing context or a closed type in a specific type theory.

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Quotients, inductive types, and quotient inductive types

2021-01-01

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories … This paper introduces an expressive class of indexed quotient-inductive types, called QWI types, within the framework of constructive type theory. They are initial algebras for indexed families of equational theories with possibly infinitary operators and equations. We prove that QWI types can be derived from quotient types and inductive types in the type theory of toposes with natural number object and universes, provided those universes satisfy the Weakly Initial Set of Covers (WISC) axiom. We do so by constructing QWI types as colimits of a family of approximations to them defined by well-founded recursion over a suitable notion of size, whose definition involves the WISC axiom. We developed the proof and checked it using the Agda theorem prover.

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Quotients, inductive types, and quotient inductive types

2022-06-07

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

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Signatures and Induction Principles for Higher Inductive-Inductive Types

2019-01-01

Ambrus Kaposi , András Kovács

Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed … Higher inductive-inductive types (HIITs) generalize inductive types of dependent type theories in two ways. On the one hand they allow the simultaneous definition of multiple sorts that can be indexed over each other. On the other hand they support equality constructors, thus generalizing higher inductive types of homotopy type theory. Examples that make use of both features are the Cauchy real numbers and the well-typed syntax of type theory where conversion rules are given as equality constructors. In this paper we propose a general definition of HIITs using a small type theory, named the theory of signatures. A context in this theory encodes a HIIT by listing the constructors. We also compute notions of induction and recursion for HIITs, by using variants of syntactic logical relation translations. Building full categorical semantics and constructing initial algebras is left for future work. The theory of HIIT signatures was formalised in Agda together with the syntactic translations. We also provide a Haskell implementation, which takes signatures as input and outputs translation results as valid Agda code.

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Relative induction principles for type theories

2021-01-01

Rafaël Bocquet , Ambrus Kaposi , Christian Sattler

We present new induction principles for the syntax of dependent type theories, which we call relative induction principles. The result of the induction principle relative to a functor F into … We present new induction principles for the syntax of dependent type theories, which we call relative induction principles. The result of the induction principle relative to a functor F into the syntax is stable over the codomain of F. We rely on the internal language of presheaf categories. In order to combine the internal languages of multiple presheaf categories, we use Dependent Right Adjoints and Multimodal Type Theory. Categorical gluing is used to prove these induction principles, but it not visible in their statements, which involve a notion of model without context extensions. As example applications of these induction principles, we give short and boilerplate-free proofs of canonicity and normalization for some small type theories, and sketch proofs of other metatheoretic results.

Relative induction principles for type theories

2021-02-23

Rafaël Bocquet , Ambrus Kaposi , Christian Sattler

Elaborating Inductive Definitions

2012-01-01

Pierre-Evariste Dagand , Conor McBride

We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating … We present an elaboration of inductive definitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive families within type theory. By elaborating an inductive definition -- a syntactic artifact -- to its code -- its semantics -- we obtain an internalized account of inductives inside the type theory itself: we claim that reasoning about inductive definitions could be carried in the type theory, not in the meta-theory as it is usually the case. Besides, we give a formal specification of that elaboration process. It is therefore amenable to formal reasoning too. We prove the soundness of our translation and hint at its correctness with respect to Coq's Inductive definitions. The practical benefits of this approach are numerous. For the type theorist, this is a small step toward bootstrapping, ie. implementing the inductive fragment in the type theory itself. For the programmer, this means better support for generic programming: we shall present a lightweight deriving mechanism, entirely definable by the programmer and therefore not requiring any extension to the type theory.

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Initial Semantics for higher-order typed syntax in Coq

2010-01-01

Benedikt Ahrens , Julianna Zsidó

Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with … Initial Semantics aims at characterizing the syntax associated to a signature as the initial object of some category. We present an initial semantics result for typed higher-order syntax together with its formalization in the Coq proof assistant. The main theorem was first proved on paper in the second author's PhD thesis in 2010, and verified formally shortly afterwards. To a simply-typed binding signature S over a fixed set T of object types we associate a category called the category of representations of S. We show that this category has an initial object Sigma(S). From its construction it will be clear that the object Sigma(S) merits the name abstract syntax associated to S. Our theorem is implemented and proved correct in the proof assistant Coq through heavy use of dependent types. The approach through monads gives rise to an implementation of syntax where both terms and variables are intrinsically typed, i.e. where the object types are reflected in the meta-level types. This article is to be seen as a research article rather than about the formalization of a classical mathematical result. The nature of our theorem - involving lengthy, technical proofs and complicated algebraic structures - makes it particularly interesting for formal verification. Our goal is to promote the use of computer theorem provers as research tools, and, accordingly, a new way of publishing mathematical results: a parallel description of a theorem and its formalization should allow the verification of correct transcription of definitions and statements into the proof assistant, and straightforward but technical proofs should be well-hidden in a digital library. We argue that Coq's rich type theory, combined with its various features such as implicit arguments, allows a particularly readable formalization and is hence well-suited for communicating mathematics.

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Normalisation by Evaluation for Type Theory, in Type Theory

2017-10-23

Thorsten Altenkirch , Ambrus Kaposi

We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a … We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction, and every construction respects the conversion relation. NBE for simple types uses a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We prove normalisation, completeness, stability and decidability of definitional equality. Most of the constructions were formalized in Agda.

Normalisation by Evaluation for Type Theory, in Type Theory

2016-01-01

Thorsten Altenkirch , Ambrus Kaposi

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Constructing Infinitary Quotient-Inductive Types

2020-01-01

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

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 … 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 involving strictly positive occurrences of 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 definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem prover.

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Extended Initiality for Typed Abstract Syntax

2012-04-06

Benedikt Ahrens , Jan Rutten

Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of 'models', yielding induction and recursion principles for abstract syntax. Zsid\'o proves … Initial Semantics aims at interpreting the syntax associated to a signature as the initial object of some category of 'models', yielding induction and recursion principles for abstract syntax. Zsid\'o proves an initiality result for simply-typed syntax: given a signature S, the abstract syntax associated to S constitutes the initial object in a category of models of S in monads. However, the iteration principle her theorem provides only accounts for translations between two languages over a fixed set of object types. We generalize Zsid\'o's notion of model such that object types may vary, yielding a larger category, while preserving initiality of the syntax therein. Thus we obtain an extended initiality theorem for typed abstract syntax, in which translations between terms over different types can be specified via the associated category-theoretic iteration operator as an initial morphism. Our definitions ensure that translations specified via initiality are type-safe, i.e. compatible with the typing in the source and target language in the obvious sense. Our main example is given via the propositions-as-types paradigm: we specify propositions and inference rules of classical and intuitionistic propositional logics through their respective typed signatures. Afterwards we use the category--theoretic iteration operator to specify a double negation translation from the former to the latter. A second example is given by the signature of PCF. For this particular case, we formalize the theorem in the proof assistant Coq. Afterwards we specify, via the category-theoretic iteration operator, translations from PCF to the untyped lambda calculus.

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Constructing Higher Inductive Types as Groupoid Quotients

2020-05-26

Niels van der Weide

In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for … In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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Two tricks to trivialize higher-indexed families

2023-01-01

Tesla Zhang

The conventional general syntax of indexed families in dependent type theories follow the style of "constructors returning a special case", as in Agda, Lean, Idris, Coq, and probably many other … The conventional general syntax of indexed families in dependent type theories follow the style of "constructors returning a special case", as in Agda, Lean, Idris, Coq, and probably many other systems. Fording is a method to encode indexed families of this style with index-free inductive types and an identity type. There is another trick that merges interleaved higher inductive-inductive types into a single big family of types. It makes use of a small universe as the index to distinguish the original types. In this paper, we show that these two methods can trivialize some very fancy-looking indexed families with higher inductive indices (which we refer to as higher indexed families).

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A general definition of dependent type theories

2020-01-01

Andrej Bauer , Philipp G. Haselwarter , Peter LeFanu Lumsdaine

We define a general class of dependent type theories, encompassing Martin-Löf's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing … We define a general class of dependent type theories, encompassing Martin-Löf's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions to be given in reasonable generality, rather than just for specific theories. Compared to other approaches, our definition stays closer to the direct or naive reading of syntax, yielding the traditional presentations of specific theories as closely as possible. Specifically, we give three main definitions: raw type theories, a minimal setup for discussing dependently typed derivability; acceptable type theories, including extra conditions ensuring well-behavedness; and well-presented type theories, generalising how in traditional presentations, the well-behavedness of a type theory is established step by step as the type theory is built up. Following these, we show that various fundamental fitness-for-purpose metatheorems hold in this generality. Much of the present work has been formalised in the proof assistant Coq.

Elaborating inductive definitions

2013-02-03

Pierre-Évariste Dagand , Conor McBride

We present an elaboration of inductive de nitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive types within type theory. By … We present an elaboration of inductive de nitions down to a universe of datatypes. The universe of datatypes is an internal presentation of strictly positive types within type theory. By elaborating an inductive de nition { a syntactic artefact { to its code { its semantics { we obtain an internalised account of inductives inside the type theory itself: we claim that reasoning about inductive de nitions could be carried in the type theory, not in the meta-theory as it is usually the case. Besides, we give a formal speci cation of that elaboration process. It is therefore amenable to formal reasoning too. We prove the soundness of our translation and hint at its completeness with respect to Coq's Inductive de nitions. The practical bene ts of this approach are numerous. For the type theorist, this is a small step toward bootstrapping, i.e. implementing the inductive fragment in the type theory itself. For the programmer, this means better support for generic programming: we shall present a lightweight deriving mechanism, entirely de nable by the programmer and therefore not requiring any extension to the type theory.

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Type Theory based on Dependent Inductive and Coinductive Types

2016-05-23

Henning Basold , Herman Geuvers

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Definitional Functoriality for Dependent (Sub)Types

2023-01-01

Théo Laurent , Meven Lennon-Bertrand , Kenji Maillard

Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional … Dependently typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws, equations satisfied propositionally by a large class of container-like type constructors $F : \mathrm{Type} \to \mathrm{Type}$, equipped with a $\mathrm{map}_{F} : (A \to B) \to F\,A \to F\,B$, such as lists or trees. Promoting these equations to definitional ones strengthens the theory, enabling slicker proofs and more automation for functorial type constructors. This extension is used to modularly justify a structural form of coercive subtyping, propagating subtyping through type formers in a map-like fashion. We show that the resulting notion of coercive subtyping, thanks to the extra definitional equations, is equivalent to a natural and implicit form of subsumptive subtyping. The key result of decidability of type-checking in a dependent type system with functor laws for lists has been entirely mechanized in Coq. This is the extended version of the work with the same name published at ESOP'24.

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Algebraic Type Theory and Universe Hierarchies

2019-01-01

Jonathan Sterling

It is commonly believed that algebraic notions of type theory support only universes à la Tarski, and that universes à la Russell must be removed by elaboration. We clarify the … It is commonly believed that algebraic notions of type theory support only universes à la Tarski, and that universes à la Russell must be removed by elaboration. We clarify the state of affairs, recalling the details of Cartmell's discipline of _generalized algebraic theory_, showing how to formulate an algebraic version of Coquand's cumulative cwfs with universes à la Russell. To demonstrate the power of algebraic techniques, we sketch a purely algebraic proof of canonicity for Martin-Löf Type Theory with universes, dependent function types, and a base type with two constants.

A Cubical Language for Bishop Sets

2022-03-29

Jonathan Sterling , Carlo Angiuli , Daniel Gratzer

We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets \`a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity … We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets \`a la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.

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Setoid Type Theory—A Syntactic Translation

2019-01-01

Thorsten Altenkirch , Simon Boulier , Ambrus Kaposi +1 more

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Parametricity and Semi-Cubical Types

2021-06-29

Hugo Moeneclaey

We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence … We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes.Our construction works not only for parametricity, but also for similar interpretations of type theory and in fact similar interpretations of any generalized algebraic theory. To be precise we consider a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory. We show that it has a right adjoint.We use techniques from locally presentable category theory, as well as from quotient inductive-inductive types.

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Quotients, inductive types, and quotient inductive types

2022-06-07

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

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Modalities and Parametric Adjoints

2022-03-29

Daniel Gratzer , Evan Cavallo , G. A. Kavvos +2 more

Birkedal et al. recently introduced dependent right adjoints as an important class of (non-fibered) modalities in type theory. We observe that several aspects of their calculus are left underdeveloped and … Birkedal et al. recently introduced dependent right adjoints as an important class of (non-fibered) modalities in type theory. We observe that several aspects of their calculus are left underdeveloped and that it cannot serve as an internal language. We resolve these problems by assuming that the modal context operator is a parametric right adjoint. We show that this hitherto unrecognized structure is common. Based on these discoveries we present a new well-behaved Fitch-style multimodal type theory, which can be used as an internal language. Finally, we apply this syntax to guarded recursion and parametricity.

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The Integers as a Higher Inductive Type

2020-05-26

Thorsten Altenkirch , Luis Scoccola

We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its … We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

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Constructing Higher Inductive Types as Groupoid Quotients

2021-04-22

Niccolò Veltri , Niels van der Weide

In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. … In this paper, we study finitary 1-truncated higher inductive types (HITs) in homotopy type theory. We start by showing that all these types can be constructed from the groupoid quotient. We define an internal notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. Then we construct a biinitial object in the bicategory of algebras in groupoids, which gives the desired algebra. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. We present several examples of HITs which are definable using our notion of signature. In particular, we show that each signature gives rise to a HIT corresponding to the freely generated algebraic structure over it. We also start the development of universal algebra in 1-types. We show that the bicategory of algebras has PIE limits, i.e. products, inserters and equifiers, and we prove a version of the first isomorphism theorem for 1-types. Finally, we give an alternative characterization of the foundamental groups of some HITs, exploiting our construction of HITs via the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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Parametricity and Semi-Cubical Types

2021-05-18

Hugo Moeneclaey

We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence … We construct a model of type theory enjoying parametricity from an arbitrary one. A type in the new model is a semi-cubical type in the old one, illustrating the correspondence between parametricity and cubes. Our construction works not only for parametricity, but also for similar interpretations of type theory and in fact similar interpretations of any generalized algebraic theory. To be precise we consider a functor forgetting unary operations and equations defining them recursively in a generalized algebraic theory. We show that it has a right adjoint. We use techniques from locally presentable category theory, as well as from quotient inductive-inductive types.

Internal Parametricity, without an Interval

2024-01-05

Thorsten Altenkirch , Yorgo Chamoun , Ambrus Kaposi +1 more

Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven … Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory with internal parametricity which is a simple extension of Martin-L\"of type theory: there are a few new type formers, term formers and equations. Geometry is not explicit in this syntax, but emergent: the new operations and equations only refer to objects up to dimension 3. We show that this theory is modelled by presheaves over the BCH cube category. Fibrancy conditions are not needed because we use span-based rather than relational parametricity. We define a gluing model for this theory implying that external parametricity and canonicity hold. The theory can be seen as a special case of a new kind of modal type theory, and it is the simplest setting in which the computational properties of higher observational type theory can be demonstrated.

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Signatures and Induction Principles for Higher Inductive-Inductive Types

2019-01-01

Ambrus Kaposi , András Kovács

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Code Generation for Higher Inductive Types

2019-01-01

Paventhan Vivekanandan

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A Note on Generalized Algebraic Theories and Categories with Families

2020-01-01

Marc Bezem , Thierry Coquand , Peter Dybjer +1 more

We give a new syntax independent definition of the notion of a generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. … We give a new syntax independent definition of the notion of a generalized algebraic theory as an initial object in a category of categories with families (cwfs) with extra structure. To this end we define inductively how to build a valid signature $Σ$ for a generalized algebraic theory and the associated category of cwfs with a $Σ$-structure and cwf-morphisms that preserve this structure on the nose. Our definition refers to uniform families of contexts, types, and terms, a purely semantic notion. Furthermore, we show how to syntactically construct initial cwfs with $Σ$-structures. This result can be viewed as a generalization of Birkhoff's completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer's construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of dependent type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a category with families.

Shallow Embedding of Type Theory is Morally Correct

2019-01-01

Ambrus Kaposi , András Kovács , Nicolai Kraus

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Generic bidirectional typing for dependent type theories

2024-01-01

Thiago Felicissimo

Abstract Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules … Abstract Bidirectional typing is a discipline in which the typing judgment is decomposed explicitly into inference and checking modes, allowing to control the flow of type information in typing rules and to specify algorithmically how they should be used. Bidirectional typing has been fruitfully studied and bidirectional systems have been developed for many type theories. However, the formal development of bidirectional typing has until now been kept confined to specific theories, with general guidelines remaining informal. In this work, we give a generic account of bidirectional typing for a general class of dependent type theories. This is done by first giving a general definition of type theories (or equivalently, a logical framework), for which we define declarative and bidirectional type systems. We then show, in a theory-independent fashion, that the two systems are equivalent. This equivalence is then explored to establish the decidability of typing for weak normalizing theories, yielding a generic type-checking algorithm that has been implemented in a prototype and used in practice with many theories.

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Gradualizing the Calculus of Inductive Constructions

2022-04-06

Meven Lennon-Bertrand , Kenji Maillard , Nicolas Tabareau +1 more

We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality … We investigate gradual variations on the Calculus of Inductive Construction (CIC) for swifter prototyping with imprecise types and terms. We observe, with a no-go theorem, a crucial tradeoff between graduality and the key properties of normalization and closure of universes under dependent product that CIC enjoys. Beyond this Fire Triangle of Graduality, we explore the gradualization of CIC with three different compromises, each relaxing one edge of the Fire Triangle. We develop a parametrized presentation of Gradual CIC (GCIC) that encompasses all three variations, and develop their metatheory. We first present a bidirectional elaboration of GCIC to a dependently-typed cast calculus, CastCIC, which elucidates the interrelation between typing, conversion, and the gradual guarantees. We use a syntactic model of CastCIC to inform the design of a safe, confluent reduction, and establish, when applicable, normalization. We study the static and dynamic gradual guarantees as well as the stronger notion of graduality with embedding-projection pairs formulated by New and Ahmed, using appropriate semantic model constructions. This work informs and paves the way towards the development of malleable proof assistants and dependently-typed programming languages.

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Multimodal Dependent Type Theory

2020-05-26

Daniel Gratzer , G. A. Kavvos , Andreas Nuyts +1 more

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We … We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion --- demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

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Decalf: A Directed, Effectful Cost-Aware Logical Framework

2023-01-01

Harrison Grodin , Yue Niu , Jonathan Sterling +1 more

We present ${\bf decalf}$, a ${\bf d}$irected, ${\bf e}$ffectful ${\bf c}$ost-${\bf a}$ware ${\bf l}$ogical ${\bf f}$ramework for studying quantitative aspects of functional programs with effects. Like ${\bf calf}$, the language … We present ${\bf decalf}$, a ${\bf d}$irected, ${\bf e}$ffectful ${\bf c}$ost-${\bf a}$ware ${\bf l}$ogical ${\bf f}$ramework for studying quantitative aspects of functional programs with effects. Like ${\bf calf}$, the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike ${\bf calf}$, the present language takes account of effects, such as probabilistic choice and mutable state; this extension requires a reformulation of ${\bf calf}$'s approach to cost accounting: rather than rely on a "separable" notion of cost, here a cost bound is simply another program. To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a recurrence that bounds the cost in a manner that readily extends to higher-order, effectful programs. The development proceeds by first introducing the ${\bf decalf}$ type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify ${\bf decalf}$ via a model in the topos of augmented simplicial sets.

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Constructing Higher Inductive Types as Groupoid Quotients

2020-05-26

Niels van der Weide

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Normalization for Multimodal Type Theory

2022-08-02

Daniel Gratzer

We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove … We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof follows from a generalization of synthetic Tait computability—an abstract approach to gluing proofs—to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.

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Constructing Infinitary Quotient-Inductive Types

2020-01-01

Marcelo Fiore , Andrew M. Pitts , S. C. Steenkamp

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Shallow Embedding of Type Theory is Morally Correct

2019-07-17

Ambrus Kaposi , András Kovács , Nicolai Kraus

There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary … There are multiple ways to formalise the metatheory of type theory. For some purposes, it is enough to consider specific models of a type theory, but sometimes it is necessary to refer to the syntax, for example in proofs of canonicity and normalisation. One option is to embed the syntax deeply, by using inductive definitions in a proof assistant. However, in this case the handling of definitional equalities becomes technically challenging. Alternatively, we can reuse conversion checking in the metatheory by shallowly embedding the object theory. In this paper, we consider the standard model of a type theoretic object theory in Agda. This model has the property that all of its equalities hold definitionally, and we can use it as a shallow embedding by building expressions from the components of this model. However, if we are to reason soundly about the syntax with this setup, we must ensure that distinguishable syntactic constructs do not become provably equal when shallowly embedded. First, we prove that shallow embedding is injective up to definitional equality, by modelling the embedding as a syntactic translation targeting the metatheory. Second, we use an implementation hiding trick to disallow illegal propositional equality proofs and constructions which do not come from the syntax. We showcase our technique with very short formalisations of canonicity and parametricity for Martin-Lof type theory. Our technique only requires features which are available in all major proof assistants based on dependent type theory.

Large and Infinitary Quotient Inductive-Inductive Types

2020-05-26

András Kovács , Ambrus Kaposi

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Multimodal Dependent Type Theory

2021-07-28

Daniel Gratzer , G. A. Kavvos , Andreas Nuyts +1 more

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We … We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

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Staged compilation with two-level type theory

2022-08-29

András Kovács

The aim of staged compilation is to enable metaprogramming in a way such that we have guarantees about the well-formedness of code output, and we can also mix together object-level … The aim of staged compilation is to enable metaprogramming in a way such that we have guarantees about the well-formedness of code output, and we can also mix together object-level and meta-level code in a concise and convenient manner. In this work, we observe that two-level type theory (2LTT), a system originally devised for the purpose of developing synthetic homotopy theory, also serves as a system for staged compilation with dependent types. 2LTT has numerous good properties for this use case: it has a concise specification, well-behaved model theory, and it supports a wide range of language features both at the object and the meta level. First, we give an overview of 2LTT's features and applications in staging. Then, we present a staging algorithm and prove its correctness. Our algorithm is "staging-by-evaluation", analogously to the technique of normalization-by-evaluation, in that staging is given by the evaluation of 2LTT syntax in a semantic domain. The staging algorithm together with its correctness constitutes a proof of strong conservativity of 2LLT over the object theory. To our knowledge, this is the first description of staged compilation which supports full dependent types and unrestricted staging for types.

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Type-Theoretic Signatures for Algebraic Theories and Inductive Types

2022-01-01

András Kovács

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Extensional concepts in intensional type theory

1995-01-01

Martin Hofmann

Conservativity of equality reflection over intensional type theory

1996-01-01

Martin Hofmann

Higher Inductive Types as Homotopy-Initial Algebras

2014-12-19

Kristina Sojakova

Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these … Homotopy Type Theory is a new field of mathematics based on the recently-discovered correspondence between Martin-Löf's constructive type theory and abstract homotopy theory. We have a powerful interplay between these disciplines - we can use geometric intuition to formulate new concepts in type theory and, conversely, use type-theoretic machinery to verify and often simplify existing mathematical proofs.

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Quotient Inductive-Inductive Types

2018-01-01

Thorsten Altenkirch , Paolo Capriotti , Gabe Dijkstra +2 more

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Displayed Categories

2017-01-01

Benedikt Ahrens , Peter LeFanu Lumsdaine

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but … We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

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Finitary Higher Inductive Types in the Groupoid Model

2018-04-01

Peter Dybjer , Hugo Moeneclaey

A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces … A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces too. We restrict attention to finitary higher inductive types and present general schemata for the types of their point, path, and surface constructors. We also derive the elimination and equality rules from the types of constructors for 1-hits and 2-hits. Moreover, we construct a groupoid model for dependent type theory with 2-hits and point out that we obtain a setoid model for dependent type theory with 1-hits by truncating the groupoid model.

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Impredicative Encodings of (Higher) Inductive Types

2018-06-27

Steve Awodey , Jonas Frey , Sam Speight

Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit … Postulating an impredicative universe in dependent type theory allows System F style encodings of finitary inductive types, but these fail to satisfy the relevant η-equalities and consequently do not admit dependent eliminators. To recover η and dependent elimination, we present a method to construct refinements of these impredicative encodings, using ideas from homotopy type theory. We then extend our method to construct impredicative encodings of some higher inductive types, such as 1-truncation and the unit circle S1.

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On Higher Inductive Types in Cubical Type Theory

2018-06-27

Thierry Coquand , Simon Huber , Anders Mörtberg

Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly … Cubical type theory provides a constructive justification to certain aspects of homotopy type theory such as Voevodsky's univalence axiom. This makes many extensionality principles, like function and propositional extensionality, directly provable in the theory. This paper describes a constructive semantics, expressed in a presheaf topos with suitable structure inspired by cubical sets, of some higher inductive types. It also extends cubical type theory by a syntax for the higher inductive types of spheres, torus, suspensions, truncations, and pushouts. All of these types are justified by the semantics and have judgmental computation rules for all constructors, including the higher dimensional ones, and the universes are closed under these type formers.

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Univalent higher categories via complete Semi-Segal types

2017-12-27

Paolo Capriotti , Nicolai Kraus

Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as … Category theory in homotopy type theory is intricate as categorical laws can only be stated "up to homotopy", and thus require coherences. The established notion of a univalent category (as introduced by Ahrens et al.) solves this by considering only truncated types, roughly corresponding to an ordinary category. This fails to capture many naturally occurring structures, stemming from the fact that the naturally occurring structures in homotopy type theory are not ordinary, but rather higher categories. Out of the large variety of approaches to higher category theory that mathematicians have proposed, we believe that, for type theory, the simplicial strategy is best suited. Work by Lurie and Harpaz motivates the following definition. Given the first ( n +3) levels of a semisimplicial type S , we can equip S with three properties: first, contractibility of the types of certain horn fillers; second, a completeness property; and third, a truncation condition. We call this a complete semi-Segal n -type . This is very similar to an earlier suggestion by Schreiber. The definition of a univalent (1-) category by Ahrens et al. can easily be extended or restricted to the definition of a univalent n -category (more precisely, ( n ,1)-category) for n ∈ {0,1,2}, and we show that the type of complete semi-Segal n -types is equivalent to the type of univalent n -categories in these cases. Thus, we believe that the notion of a complete semi-Segal n -type can be taken as the definition of a univalent n -category. We provide a formalisation in the proof assistant Agda using a completely explicit representation of semi-simplicial types for levels up to 4.

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Semantics of higher inductive types

2019-06-17

Peter LeFanu Lumsdaine , Michael Shulman

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development … Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the "synthetic" development of homotopy theory within type theory, as well as in formalizing ordinary set-level mathematics in type theory. In this article, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has *weakly stable typal initial algebras* for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specializes to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction, and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed $(\infty,1)$-category is presented by some model category to which our results apply.

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The biequivalence of locally cartesian closed categories and Martin-Löf type theories

2014-04-29

Pierre Clairambault , Peter Dybjer

Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category … Seely's paper "Locally cartesian closed categories and type theory" contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-L\"of type theories with Pi-types, Sigma-types and extensional identity types. However, Seely's proof relies on the problematic assumption that substitution in types can be interpreted by pullbacks. Here we prove a corrected version of Seely's theorem: that the B\'enabou-Hofmann interpretation of Martin-L\"of type theory in locally cartesian closed categories yields a biequivalence of 2-categories. To facilitate the technical development we employ categories with families as a substitute for syntactic Martin-L\"of type theories. As a second result we prove that if we remove Pi-types the resulting categories with families are biequivalent to left exact categories.

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Cubical Type Theory: a constructive interpretation of the univalence axiom

2015-05-18

Cyril Cohen , Thierry Coquand , Simon Huber +1 more

This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in … This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky's univalence axiom is provable in this system. We also explain an extension with some higher inductive types like the circle and propositional truncation. Finally we provide semantics for this cubical type theory in a constructive meta-theory.

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Type Theory based on Dependent Inductive and Coinductive Types

2016-07-05

Henning Basold , Herman Geuvers

We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a … We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For example, all well-known basic types and type formers that are needed for using this type theory as a logic are definable: propositional connectives, like falsity, conjunction, disjunction, and function space, dependent function space, existential quantification, equality, natural numbers, vectors etc. The reduction relation on terms consists solely of a rule for recursion and a rule for corecursion. The reduction relations for well-known types arise from that. To further support the introduction of this new type theory, we also prove fundamental properties of its term calculus. Most importantly, we prove subject reduction and strong normalisation of the reduction relation, which gives computational meaning to the terms.

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