Eliminating dependent pattern matching without K

Cockx, Jesper; Devriese, Dominique; Piessens, Frank

doi:10.1017/s0956796816000174

Cited by 7 publications

(6 citation statements)

References 23 publications

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“…It can be seen as a generic method of constructing an inversion principle (McBride, 1998b). It is also a core component in the translation of definitions by pattern matching to eliminators (Goguen et al, 2006;Cockx, Devriese, and Piessens, 2016b).…”

Section: Specialization By Unificationmentioning

confidence: 99%

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Proof-relevant unification: Dependent pattern matching with only the axioms of your type theory

Cockx

Devriese

2018

J. Funct. Prog.

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Dependently typed languages such as Agda, Coq, and Idris use a syntactic first-order unification algorithm to check definitions by dependent pattern matching. However, standard unification algorithms implicitly rely on principles such as uniqueness of identity proofs and injectivity of type constructors. These principles are inadmissible in many type theories, particularly in the new and promising branch known as homotopy type theory. As a result, programs and proofs in these new theories cannot make use of dependent pattern matching or other techniques relying on unification, and are as a result much harder to write, modify, and understand. This paper proposes a proof-relevant framework for reasoning formally about unification in a dependently typed setting. In this framework, unification rules compute not just a unifier but also a corresponding soundness proof in the form of an equivalence between two sets of equations. By rephrasing the standard unification rules in a proof-relevant manner, they are guaranteed to preserve soundness of the theory. In addition, it enables us to safely add new rules that can exploit the dependencies between the types of equations, such as rules for eta-equality of record types and higher dimensional unification rules for solving equations between equality proofs. Using our framework, we implemented a complete overhaul of the unification algorithm used by Agda. As a result, we were able to replace previous ad-hoc restrictions with formally verified unification rules, fixing a substantial number of bugs in the process. In the future, we may also want to integrate new principles with pattern matching, for example, the higher inductive types introduced by homotopy type theory. Our framework also provides a solid basis for such extensions to be built on.

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Section: Specialization By Unificationmentioning

confidence: 99%

“…But as a term in type theory, it also has a certain computational behaviour. This computational behaviour is important for the applications we have in mind, in particular, the translation of pattern matching to eliminators (Cockx et al, 2016b).…”

Section: Computational Behaviour Of Unification Rulesmentioning

confidence: 99%

Proof-relevant unification: Dependent pattern matching with only the axioms of your type theory

Cockx

Devriese

2018

J. Funct. Prog.

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“…This should come as no surprise to implementors of pattern-matching compilers (McBride, 2000;Sozeau, 2010;Cockx et al, 2016) who had, time and again, to fight a struggle between generalization of dependent types -which introduces equalities -and simplification of the resulting terms -which collapses as many equalities as possible. These difficulties motivated in a large part the adoption of UIP for it provides further opportunities for simplification thus making the resulting terms manageable.…”

Section: A Homotopical Detourmentioning

confidence: 99%

Foundations of dependent interoperability

Dagand¹,

Tabareau²,

Tanter³

2018

J. Funct. Prog.

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Full-spectrum dependent types promise to enable the development of correct-by-construction software. However, even certified software needs to interact with simply-typed or untyped programs, be it to perform system calls, or to use legacy libraries. Trading static guarantees for runtime checks, the dependent interoperability framework provides a mechanism by which simply-typed values can safely be coerced to dependent types and, conversely, dependently-typed programs can defensively be exported to a simply-typed application. In this article, we give a semantic account of dependent interoperability. Our presentation relies on and is guided by a pervading notion of type equivalence, whose importance has been emphasized in recent work on homotopy type theory. Specifically, we develop the notions of type-theoretic partial Galois connections as a key foundation for dependent interoperability, which accounts for the partiality of the coercions between types. We explore the applicability of both type-theoretic Galois connections and anticonnections in the setting of dependent interoperability. A partial Galois connection enforces a translation of dependent types to runtime checks that are both sound and complete with respect to the invariants encoded by dependent types. Conversely, picking an anticonnection instead lets us induce weaker, sound conditions that can amount to more efficient runtime checks.Our framework is developed in Coq; it is thus constructive and verified in the strictest sense of the terms. Using our library, users can specify domain-specific partial connections between data structures. Our library then takes care of the (sometimes, heavy) lifting that leads to interoperable programs. It thus becomes possible, as we shall illustrate, to internalize and hand-tune the extraction of dependently-typed programs to interoperable OCaml programs within Coq itself.

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“…In the implementation, the formalisation is type-checked using the flag without-K for compatibility with HoTT [4]. Also, the flag exact-split was used to ensure that all clauses in a definition are definitional equalities.…”

Section: Introductionmentioning

confidence: 99%

On homotopy of walks and spherical maps in homotopy type theory

Prieto-Cubides

2022

Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs

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We work with combinatorial maps to represent graph embeddings into surfaces up to isotopy. The surface in which the graph is embedded is left implicit in this approach. The constructions herein are proof-relevant and stated with a subset of the language of homotopy type theory.This article presents a refinement of one characterisation of embeddings in the sphere, called spherical maps, of connected and directed multigraphs with discrete node sets. A combinatorial notion of homotopy for walks and the normal form of walks under a reduction relation is introduced. The first characterisation of spherical maps states that a graph can be embedded in the sphere if any pair of walks with the same endpoints are merely walk-homotopic. The refinement of this definition filters out any walk with inner cycles. As we prove in one of the lemmas, if a spherical map is given for a graph with a discrete node set, then any walk in the graph is merely walk-homotopic to a normal form.The proof assistant Agda contributed to formalising the results recorded in this article.

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Eliminating dependent pattern matching without K

Cited by 7 publications

References 23 publications

Proof-relevant unification: Dependent pattern matching with only the axioms of your type theory

Proof-relevant unification: Dependent pattern matching with only the axioms of your type theory

Foundations of dependent interoperability

On homotopy of walks and spherical maps in homotopy type theory

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