Finite sets in homotopy type theory

Frumin, Dan; Geuvers, Herman; Gondelman, Léon; Weide, Niels van der

doi:10.1145/3167085

Cited by 14 publications

(10 citation statements)

References 40 publications

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“…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%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2021

Preprint

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We develop a constructive theory of finite multisets, defining them as free commutative monoids in Homotopy Type Theory. We formalise two algebraic presentations of this construction using 1-HITs, establishing the categorical universal property for each and thereby their equivalence. These presentations correspond to equational theories including a commutation axiom. In this setting, we prove important structural combinatorial properties of singleton multisets arising from concatenations and projections of multisets. This is done in generality, without assuming decidable equality on the carrier set. Further, as applications, we present a constructive formalisation of the relational model of differential linear logic and use it to characterise the equality type of multisets. This leads us to a novel conditional equational presentation of the finite-multiset construction and thereby to a sound and complete deduction system for multiset equality.

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“…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%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2021

Preprint

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“…There are several different ways to describe finite sets and finite subsets in type theory [Spiwack and Coquand 2010]. Recently, Frumin et al [2018] have presented an implementation of the finite powerset functor in HoTT as a higher inductive type. Given a type A, they construct the type P fin A of finite subsets of A as the free join semilattice over A.…”

Section: The Finite Powerset Functormentioning

confidence: 99%

“…The type P fin A comes with a rather complex induction principle, which is similar to the one described by Frumin et al [2018]. We spell it out for the case in which the type we are eliminating into is a proposition.…”

Section: :11mentioning

confidence: 99%

“…Using the induction principle of the finite powerset construction, it is also possible to prove an extensionality principle for finite subsets: two subsets x, y : P fin A are path equal if and only if they contain the same elements, i.e. if the type Πa : A. a ∈ x ↔ a ∈ y is inhabited [Frumin et al 2018].…”

Section: :11mentioning

confidence: 99%

“…As a running example, we study guarded labelled transition systems (GLTSs), i.e., coalgebras for the functor P fin (A × ▷(−)). Here P fin is the finite powerset functor, which can be defined [Frumin et al 2018] as a higher inductive type (HIT) [Univalent Foundations Program 2013], i.e., an inductive type with constructors for elements as well as for equalities. We show that path equality in the final coalgebra for P fin (A × ▷(−)) coincides with a natural adaptation of bisimilarity for finitely branching labelled transition systems to their guarded variant.…”

Section: Introductionmentioning

confidence: 99%

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Bisimulation as path type for guarded recursive types

Møgelberg

Veltri

2019

Proc. ACM Program. Lang.

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In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, allowing this to be encoded in types. Indeed, coinductive types can be encoded using a combination of guarded recursion and universal quantification over clocks. This paper studies the notion of bisimilarity for guarded recursive types in Ticked Cubical Type Theory, an extension of Cubical Type Theory with guarded recursion. We prove that, for any functor, an abstract, category theoretic notion of bisimilarity for the final guarded coalgebra is equivalent (in the sense of homotopy type theory) to path equality (the primitive notion of equality in cubical type theory). As a worked example we study a guarded notion of labelled transition systems, and show that, as a special case of the general theorem, path equality coincides with an adaptation of the usual notion of bisimulation for processes. In particular, this implies that guarded recursion can be used to give simple equational reasoning proofs of bisimilarity. This work should be seen as a step towards obtaining bisimilarity as path equality for coinductive types using the encodings mentioned above.

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