A Type-Checking Algorithm for Martin-Löf Type Theory with Subtyping Based on Normalisation by Evaluation

Fridlender, Daniel; Pagano, Miguel

doi:10.1007/978-3-642-38946-7_12

Cited by 2 publications

(5 citation statements)

References 14 publications

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“…The proofs follow roughly Fridlender and Pagano [2013], for details, see the long version of this article. Fig 11 defines an incremental subtyping algorithm Γ ⊢ T <: T ′ .…”

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

“…Simultaneously by induction on Γ ⊢ T ® A ∈ s. □ Lemma 5.3 (Semantic implies judgmental subtyping [Fridlender and Pagano 2013]). …”

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

“…The proof follows Abel et al [2007] and Fridlender and Pagano [2013] and defines a Kripke logical relation Γ ⊢ t : T ® f ∈ A between a well-typed term Γ ⊢ t : T and a value f ∈ A. First, let us define some auxiliary judgements that relate a well-formed syntactic object to a value, via read back.…”

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

“…In this, it improves on Fridlender and Pagano [2013] which instruments full normalization (NbE) at every step. This article is accompanied by a prototypical type checker Sit which implements the type system and type checking algorithm as described in the remainder of the paper.…”

Section: Introductionmentioning

confidence: 99%

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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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“…The proofs follow roughly Fridlender and Pagano [2013], for details, see the long version of this article. Fig 11 defines an incremental subtyping algorithm Γ ⊢ T <: T ′ .…”

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

“…Simultaneously by induction on Γ ⊢ T ® A ∈ s. □ Lemma 5.3 (Semantic implies judgmental subtyping [Fridlender and Pagano 2013]). …”

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

Section: Soundness Of Normalization By Evaluationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Normalization by evaluation for sized dependent types

Abel

Vezzosi

Winterhalter³

2017

Proc. ACM Program. Lang.

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“…Another interesting direction is to extend NbE to impredicative PTSs, Abel (2010) has made some progress towards defining NbE for the core calculus of constructions. Since NbE can also cope with systems featuring subtyping (Fridlender & Pagano, 2013), the method should also be useful for PTSs with a hierarchy of sorts as in ECC (Luo, 1994).…”

Section: Discussionmentioning

confidence: 99%

Pure type systems with explicit substitutions

Fridlender¹,

Pagano²

2015

J. Funct. Prog.

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We introduce a new formulation of pure type systems (PTSs) with explicit substitution and de Bruijn indices and formally prove some of its meta-theory. Using techniques based on Normalisation by Evaluation, we prove that untyped conversion can be typed for predicative PTSs. Although this equivalence was settled by Siles and Herbelin for the conventional presentation of PTSs, we strongly conjecture that our proof method can also be applied to PTSs with η.

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A Type-Checking Algorithm for Martin-Löf Type Theory with Subtyping Based on Normalisation by Evaluation

Cited by 2 publications

References 14 publications

Normalization by evaluation for sized dependent types

Normalization by evaluation for sized dependent types

Pure type systems with explicit substitutions

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