Linear Sized Types in the Calculus of Constructions

Sacchini, Jorge Luis

doi:10.1007/978-3-319-07151-0_11

Cited by 6 publications

(9 citation statements)

References 21 publications

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“…Well-founded ordinal induction and size change principle. In this paper, inductive and coinductive types carry an ordinal number κ to form sized types µ κ X.F (X) and ν κ X.F (X) [3,17,38]. Intuitively, they correspond to κ iterations of F on the types ⊥ and ⊤ respectively.…”

Section: ∀X(a → B) ⊂ (∀Xa) → (∀Xb)mentioning

confidence: 99%

Practical Subtyping for System F with Sized (Co-)Induction

Lepigre,

Raffalli

2016

Preprint

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We present a rich type system with subtyping for an extension of System F. Our type constructors include sum and product types, universal and existential quantifiers, inductive and coinductive types. The latter two may carry annotations allowing the encoding of size invariants that are used to ensure the termination of recursive programs. For example, the termination of quicksort can be derived by showing that partitioning a list does not increase its size. The system deals with complex programs involving mixed induction and coinduction, or even mixed polymorphism and (co-)induction (as for Scottencoded data types). One of the key ideas is to completely separate the notion of size from recursion. We do not check the termination of programs directly, but rather show that their (circular) typing proofs are well-founded. We then obtain termination using a standard semantic proof of normalisation. To demonstrate the practicality of our system, we provide an implementation which accepts all the examples discussed in the paper.

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Section: ∀X(a → B) ⊂ (∀Xa) → (∀Xb)mentioning

confidence: 99%

Practical Subtyping for System F with Sized (Co-)Induction

Lepigre,

Raffalli

2016

Preprint

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“…For more details about this kind of sized types, including normalisation proofs (in one case sketched), see Abel and Pientka [2016] and Sacchini [2015]. Note that Abel and Pientka discuss a language without dependent types (based on System F ω ), and that while Sacchini's unpublished draft treats a dependently typed language, this language differs from Agda.…”

Section: Up-to Techniques Usingmentioning

confidence: 99%

“…Some type theories support a more flexible variant of corecursion based on sized types [Hughes et al 1996;Amadio and Coupet-Grimal 1998;Giménez 1998;Xi 2002;Blanqui 2004Blanqui , 2005Barthe et al 2004Barthe et al , 2006Grégoire and Sacchini 2010;Abel 2012;Sacchini 2013Sacchini , 2014Abel and Pientka 2016;Abel et al 2017]. Sized types tend to make the type theory more complicated, but my experienceÐbased on using what is perhaps the most mature implementation of type theory with sized types, Agda [Agda Team 2017]Ðis that they make it much easier to write corecursive programs.…”

Section: Introductionmentioning

confidence: 99%

Up-to techniques using sized types

Danielsson

2017

Proc. ACM Program. Lang.

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Up-to techniques are used to make it easier—or feasible—to construct, for instance, proofs of bisimilarity. This text shows how many up-to techniques can be framed as size-preserving functions , using sized types to keep track of sizes. Through a number of examples it is argued that this approach to up-to techniques is often convenient to use in practice. Some examples of functions that cannot be made size-preserving are also included, in order to illustrate the limits of the approach. On the more theoretical side a class of up-to techniques intended to capture a natural mode of use of size-preserving functions is defined. This class turns out to correspond closely to "functions below the companion", a notion recently introduced by Pous.

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“…Type based approaches to termination add size parameters to type system as a means to guarantee that recursive functions terminate. The typing rule L F illustrates a (simpli ed) type-based approach to using size variables in recursive de nitions (adapted from [10,90]):…”

Section: Introductionmentioning

confidence: 99%

“…Other type systems involve more complex size algebras. For example, a more expressive language using linear sized types was introduced in [90] by extending the Calculus of Inductive Constructions with (co-)inductive types and size annotations. Other systems introduce sizes as upper bounds [9,2], or add sized types in a dependently typed framework with polymorphism and indexed types [111].…”

Section: Introductionmentioning

confidence: 99%

Mechanizing the metatheory of rewire

Reynolds¹

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The [lambda]-calculus provides a simple, well-established framework for research in functional programming languages that readily lends itself to the use offormal methods--that is, the use of mathematically sound techniques and supporting tools--to describe and verify properties of programming languages, as well. This is no coincidence. After all, the [lambda]-calculus formalizes the concept of effective computability, for all computable functions are definable in the untyped [lambda]-calculus, making it expressively equivalent torecursive functions. In software, the expressiveness of functional languages is considereda strength. Functional approaches to language design, however, needn't be limited to soft-ware. In hardware, the expressiveness of functional languages becomes a major obstacle to successful hardware synthesis, for the reason that such languages are usually capable of expressing general recursion. The presence of general recursion makes it possible to generate expressions that run forever, never producing a well-defined value. In this dissertation, we study two novel variants of the simply typed [lambda]-calculus, representing fragments of functional hardware description languages. The first variant extends the type system, using natural numbers representing time. This addition, though simple, is non-trivial. We prove that this calculus possesses bounded variants of type-safety and strong normalization. That is to say, we show that all well-typed expressions evaluate to values within a bound determined by the natural number index of their corresponding types. The second variant is a computational [lambda]-calculus that formalizes the core fragment of the hardware description language known as ReWire. We prove that the language has type-safety and is strongly normalizing -- the proof of strong normalizationis the first mechanized proof of its kind. We define an equational theory with respect to this language. This allows us to prove that the language has desirable security properties by construction. This work supports a full-edged, formal methodology for producing high assurance hardware.

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Linear Sized Types in the Calculus of Constructions

Cited by 6 publications

References 21 publications

Practical Subtyping for System F with Sized (Co-)Induction

Practical Subtyping for System F with Sized (Co-)Induction

Up-to techniques using sized types

Mechanizing the metatheory of rewire

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