Towards Normalization by Evaluation for the βη-Calculus of Constructions

Abel, Andreas

doi:10.1007/978-3-642-12251-4_17

Cited by 7 publications

(4 citation statements)

References 19 publications

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“…NbE has proven to be a robust method to decide equality in powerful type theories with non-trivial η-laws. It scales to universes and large eliminations [Abel et al 2007], topped with singleton types or proof irrelevance [Abel et al 2011], and even impredicativity [Abel 2010]. At its heart there are reflection ↑ T and reification ↓ T functions directed by type T and orchestrating just-in-time η-expansion.…”

Section: Introductionmentioning

confidence: 99%

Normalization by evaluation for sized dependent types

Abel

Vezzosi

Winterhalter³

2017

Proc. ACM Program. Lang.

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Sized types have been developed to make termination checking more perspicuous, more powerful, and more modular by integrating termination into type checking. In dependently-typed proof assistants where proofs by induction are just recursive functional programs, the termination checker is an integral component of the trusted core, as validity of proofs depend on termination. However, a rigorous integration of full-fledged sized types into dependent type theory is lacking so far. Such an integration is non-trivial, as explicit sizes in proof terms might get in the way of equality checking, making terms appear distinct that should have the same semantics.In this article, we integrate dependent types and sized types with higher-rank size polymorphism, which is essential for generic programming and abstraction. We introduce a size quantifier ∀ which lets us ignore sizes in terms for equality checking, alongside with a second quantifier Π for abstracting over sizes that do affect the semantics of types and terms. Judgmental equality is decided by an adaptation of normalization-by-evaluation for our new type theory, which features type shape-directed reflection and reification. It follows that subtyping and type checking of normal forms are decidable as well, the latter by a bidirectional algorithm.

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Section: Introductionmentioning

confidence: 99%

Normalization by evaluation for sized dependent types

Abel

Vezzosi

Winterhalter³

2017

Proc. ACM Program. Lang.

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“…It depends on a substitution (for which the predicate needs to hold as well) and a term. P AΨ (ρ, s, α) expresses that the logical predicate 2 For reference, the "arguments for the eliminator" notation for the motives looks as follows.…”

Section: The Logical Predicate Interpretationmentioning

confidence: 99%

“…Danvy is using semantic normalisation for partial evaluation [20]. Normalisation by evaluation using untyped realizers has been applied to dependent types by Abel et al [2][3][4]. Danielsson [19] has formalized NBE for dependent types but he doesn't prove soundness of normalisation.…”

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confidence: 99%

Normalisation by Evaluation for Type Theory, in Type Theory

Altenkirch¹,

Kaposi²

2017

Logical Methods in Computer Science ; Volume 13

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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.

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“…Both presentations have their own purpose but in two different directions. Because they carry more typing information, the systems based on judgmental equality are convenient for building models (Goguen, 1994;Abel et al, 2007;Abel, 2010;Werner & Lee, 2011). On the other hand the typing judgments are irrelevant for computation, and with untyped conversion one can concentrate on the purely computational content of conversion.…”

Section: Introductionmentioning

confidence: 99%

Pure Type System conversion is always typable

Siles¹,

Herbelin²

2012

J. Funct. Prog.

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AbstractPure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step towards this equivalence has been made by Adams for a particular class ofPure Type Systems(PTS) called functional. Then, his result has been relaxed to all semi-full PTSs in previous work. In this paper, we finally give a positive answer to the general question, and prove that equivalence holds for any Pure Type System.

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Towards Normalization by Evaluation for the βη-Calculus of Constructions

Cited by 7 publications

References 19 publications

Normalization by evaluation for sized dependent types

Normalization by evaluation for sized dependent types

Normalisation by Evaluation for Type Theory, in Type Theory

Pure Type System conversion is always typable

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