Analysis of a guard condition in type theory

Amadio, Roberto M.; Coupet-Grimal, Solange

doi:10.1007/bfb0053541

Cited by 22 publications

(25 citation statements)

References 11 publications

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“…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%

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: Introductionmentioning

confidence: 99%

Wellfounded recursion with copatterns

Abel

Pientka

2013

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

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“…This idea has been taken up by Hughes, Pareto, and Sabry [17], Giménez [16], Amadio and Coupet-Grimal [9], Barthe et al [10], Blanqui [14] and myself [1,2]. My thesis [5] describes F ω , an extension of the higher-order polymorphic lambda-calculus F ω by sized inductive types and structural recursion.…”

Section: Introductionmentioning

confidence: 99%

Implementing a normalizer using sized heterogeneous types

Abel

2009

J. Funct. Prog.

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In the simply-typed lambda-calculus, a hereditary substitution replaces a free variable in a normal form r by another normal form s of type a, removing freshly created redexes on the fly. It can be defined by lexicographic induction on a and r, thus, giving rise to a structurally recursive normalizer for the simply-typed lambda-calculus. We generalize this scheme to simultaneous substitutions, preserving its simple termination argument. We further implement hereditary simultaneous substitutions in a functional programming language with sized heterogeneous inductive types, Fω b, arriving at an interpreter whose termination can be tracked by the type system of its host programming language.

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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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Analysis of a guard condition in type theory

Cited by 22 publications

References 11 publications

Wellfounded recursion with copatterns

Wellfounded recursion with copatterns

Implementing a normalizer using sized heterogeneous types

Up-to techniques using sized types

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