A Modular Type-checking algorithm for Type Theory with Singleton Types and Proof Irrelevance

Abel, Andreas; Coquand, Thierry; Pagano, Miguel

doi:10.2168/lmcs-7(2:4)2011

Cited by 8 publications

(9 citation statements)

References 44 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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“…In [29], a judgement "A is provable" is introduced, to say that a proof of A exists, but no attention is paid to what it is. Similarly, [1] introduces proof irrelevance in Martin-Löf's logical framework using a function to distinguish propositions A from "proof-irrelevant propositions Prf(A)". While A can be inhabited by several normal terms, Prf(A) is inhabited by only one normal form noted ⋆, to which all terms of Prf(A) reduce.…”

Section: Related Workmentioning

confidence: 99%

Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory

Hondet¹,

Blanqui²

2021

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The λΠ-calculus modulo theory is a logical framework in which various logics and type systems can be encoded, thus helping the cross-verification and interoperability of proof systems based on those logics and type systems. In this paper, we show how to encode predicate subtyping and proof irrelevance, two important features of the PVS proof assistant. We prove that this encoding is correct and that encoded proofs can be mechanically checked by Dedukti, a type checker for the λΠ-calculus modulo theory using rewriting.

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“…To bring out the ideas underlying our approach without too much syntactic clutter, we use the familiar and simple example of such data, the terms of the untyped -calculus [2] Λ ≜ { ::= | | . } (1) where ranges over an infinite set of variables and where terms are identified up to the usual notion of -equivalence (≡ ) forbound variables.…”

Section: Introductionmentioning

confidence: 99%

“…For example, the function (−) = (−)[ ′ / ′ ] : Λ → Λ for capture-avoiding substitution of ′ ∈ Λ for ′ ∈ is given by taking 1 3 ≜ . and to be the finite set consisting of ′ and the free variables of ′ .…”

Section: Introductionmentioning

confidence: 99%

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Nominal system T

Pitts

2010

Proceedings of the 37th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

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This paper introduces a new recursion principle for inductive data modulo -equivalence of bound names. It makes use of Oderskystyle local names when recursing over bound names. It is formulated in an extension of Gödel's System T with names that can be tested for equality, explicitly swapped in expressions and restricted to a lexical scope. The new recursion principle is motivated by the nominal sets notion of " -structural recursion", whose use of names and associated freshness side-conditions in recursive definitions formalizes common practice with binders. The new Nominal System T presented here provides a calculus of total functions that is shown to adequately represent -structural recursion while avoiding the need to verify freshness side-conditions in definitions and computations. Adequacy is proved via a normalizationby-evaluation argument that makes use of a new semantics of local names in Gabbay-Pitts nominal sets.

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A Modular Type-checking algorithm for Type Theory with Singleton Types and Proof Irrelevance

Cited by 8 publications

References 44 publications

Normalization by evaluation for sized dependent types

Normalization by evaluation for sized dependent types

Encoding of Predicate Subtyping with Proof Irrelevance in the λΠ-Calculus Modulo Theory

Nominal system T

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