Type-Based Productivity of Stream Definitions in the Calculus of Constructions

Sacchini, Jorge Luis

doi:10.1109/lics.2013.29

Cited by 24 publications

(35 citation statements)

References 13 publications

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“…Previous approaches to type-based termination (Hughes et al 1996;Amadio and Coupet-Grimal 1998;Barthe et al 2004;Blanqui 2004;Sacchini 2013) have defined approximants of least µ α F and greatest fixed-points ν α F of monotone type constructors F ∈ CR + → CR by conventional induction on ordinal α, distinguishing zero (0), successor (α + 1), and limit ordinals (λ).…”

Section: Ordinals and Fixed-pointsmentioning

confidence: 99%

“…Both issues are crucial since we want to program inductive proofs as recursive functions and coinductive proofs as infinite objects or corecursive functions producing infinite objects. In this article, we adapt typebased termination (Hughes et al 1996;Amadio and Coupet-Grimal 1998;Barthe et al 2004;Blanqui 2004;Abel 2008b;Sacchini 2013) to definitions by copatterns.…”

Section: Introductionmentioning

confidence: 99%

“…We are the first to utilize inflationary iteration in a type system. • Well-founded induction alleviates the need for a semi-continuity check for sized types of recursive functions (Hughes et al 1996;Abel 2008b) which sometimes disguises itself as a monotonicity check (Barthe et al 2004;Blanqui 2004;Barthe et al 2008;Sacchini 2013). Thus, we put type-based termination on leaner and better understandable foundations.…”

Section: Introductionmentioning

confidence: 99%

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Wellfounded recursion with copatterns

Abel

Pientka

2013

Proceedings of the 18th ACM SIGPLAN International Conference on Functional Programming

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In this paper, we study strong normalization of a core language based on System Fω which supports programming with finite and infinite structures. Building on our prior work, finite data such as finite lists and trees are defined via constructors and manipulated via pattern matching, while infinite data such as streams and infinite trees is defined by observations and synthesized via copattern matching. In this work, we take a type-based approach to strong normalization by tracking size information about finite and infinite data in the type. This guarantees compositionality. More importantly, the duality of pattern and copatterns provide a unifying semantic concept which allows us for the first time to elegantly and uniformly support both well-founded induction and coinduction by mere rewriting. The strong normalization proof is structured around Girard's reducibility candidates. As such our system allows for non-determinism and does not rely on coverage. Since System Fω is general enough that it can be the target of compilation for the Calculus of Constructions, this work is a significant step towards representing observation-centric infinite data in proof assistants such as Coq and Agda.

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Section: Ordinals and Fixed-pointsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Wellfounded recursion with copatterns

Abel

Pientka

2013

Proceedings of the 18th ACM SIGPLAN International Conference on Functional Programming

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“…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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“…Since elements of coinductive types are sometimes called codata, we refer to this approach as productive coprogramming via guarded recursion. Guarded recursion can be seen as a light-weight alternative to sized types [1,18].…”

Section: Introductionmentioning

confidence: 99%

A type theory for productive coprogramming via guarded recursion

Møgelberg

2014

Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Nin

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To ensure consistency and decidability of type checking, proof assistants impose a requirement of productivity on corecursive definitions. In this paper we investigate a type-based alternative to the existing syntactic productivity checks of Coq and Agda, using a combination of guarded recursion and quantification over clocks. This approach was developed by Atkey and McBride in the simply typed setting, here we extend it to a calculus with dependent types. Building on previous work on the topos-of-trees model we construct a model of the calculus using a family of presheaf toposes, each of which can be seen as a multi-dimensional version of the topos-of-trees. As part of the model construction we must solve the coherence problem for modelling dependent types in locally cartesian closed categories simulatiously in a whole family of locally cartesian closed categories. We do this by embedding all the categories in a large one and applying a recent approach to the coherence problem due to Streicher and Voevodsky.

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Type-Based Productivity of Stream Definitions in the Calculus of Constructions

Cited by 24 publications

References 13 publications

Wellfounded recursion with copatterns

Wellfounded recursion with copatterns

Up-to techniques using sized types

A type theory for productive coprogramming via guarded recursion

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