Bag Equivalence via a Proof-Relevant Membership Relation

Danielsson, Nils Anders

doi:10.1007/978-3-642-32347-8_11

Cited by 5 publications

(3 citation statements)

References 15 publications

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“…The simplest encoding of finite multisets is lists up to permutation of their elements. In this context, the relation of Definition 6.10 using automorphisms on finite cardinals as permutations has been considered by Altenkirch et al [1] and Nuo [41]; while Danielsson [20] has considered a similar relation for bag equivalence of lists. Multisets in type theory have also been studied by Gylterud [31] from the point of view of constructive foundations, considering how to generalise set-theoretic axioms from sets to multisets.…”

Section: Finite Multisetsmentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space.

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Section: Finite Multisetsmentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…These transports are done in an ad-hoc way that can be done more elegantly with the SIP. Prior to HoTT/UF, Danielsson [2012] formalized finite multisets as a setoid of lists modulo "bag equivalence, " which is exactly the relation R that we use to quotient lists in Example 5.5.…”

Section: Related and Future Workmentioning

confidence: 99%

Internalizing representation independence with univalence

Angiuli

Cavallo

Mörtberg

et al. 2021

Proc. ACM Program. Lang.

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In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations by transferring them from simpler, related implementations. Recent work in proof assistants has shown that Voevodsky's univalence principle allows transferring theorems between isomorphic types, but many instances of representation independence in programming involve non-isomorphic representations. In this paper, we develop techniques for establishing internal relational representation independence results in dependent type theory, by using higher inductive types to simultaneously quotient two related implementation types by a heterogeneous correspondence between them. The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence. We illustrate our techniques by considering applications to matrices, queues, and finite multisets. Our results are all formalized in Cubical Agda, a recent extension of Agda which supports univalence and higher inductive types in a computationally well-behaved way.

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“…Other definiitons of finite multisets in type theory have been considered before, for instance using setoids [15,26,4]. None of these encodings prove the universal property of free commutative monoids and most results use the assumption of decidable equality.…”

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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Bag Equivalence via a Proof-Relevant Membership Relation

Cited by 5 publications

References 15 publications

Free Commutative Monoids in Homotopy Type Theory

Free Commutative Monoids in Homotopy Type Theory

Internalizing representation independence with univalence

Free Commutative Monoids in Homotopy Type Theory

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