Size-based termination of higher-order rewriting

Blanqui, Frédéric

doi:10.1017/s0956796818000072

Cited by 4 publications

(2 citation statements)

References 149 publications

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“…But we may want to go further because the structural ordering is not enough to handle the following system which is not accepted by Agda: A solution to handle this system is to use arguments filterings (remove the second argument of -) or simple projections [17]. Another one is to extend the type system with size annotations as in Agda and compute the SCT matrices by comparing the size of terms instead of their structure [1,7]. In our example, the size of m -n is smaller than or equal to the size of m. One can deduce this by using user annotations like in Agda, or by using heuristics [8].…”

Section: Discussionmentioning

confidence: 99%

Dependency Pairs Termination in Dependent Type Theory Modulo Rewriting

Blanqui,

Genestier,

Hermant

2019

Preprint

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Dependency pairs are a key concept at the core of modern automated termination provers for first-order term rewriting systems. In this paper, we introduce an extension of this technique for a large class of dependently-typed higher-order rewriting systems. This extends previous results by Wahlstedt on the one hand and the first author on the other hand to strong normalization and non-orthogonal rewriting systems. This new criterion is implemented in the type-checker Dedukti.

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Section: Discussionmentioning

confidence: 99%

Dependency Pairs Termination in Dependent Type Theory Modulo Rewriting

Blanqui,

Genestier,

Hermant

2019

Preprint

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“…It has been extended to handle dependent types by Jouannaud and Li [2015]. The size-change termination principle [Lee et al 2001] has been extended to dependently typed rewriting systems by Blanqui [2004Blanqui [ , 2018 and has recently been implemented in the SizeChangeTool [Genestier 2019] to check termination of rewrite rules in Dedukti.…”

Section: Extensions and Future Workmentioning

confidence: 99%

The taming of the rew: a type theory with computational assumptions

Cockx

Tabareau

Winterhalter

2021

Proc. ACM Program. Lang.

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Dependently typed programming languages and proof assistants such as Agda and Coq rely on computation to automatically simplify expressions during type checking. To overcome the lack of certain programming primitives or logical principles in those systems, it is common to appeal to axioms to postulate their existence. However, one can only postulate the bare existence of an axiom, not its computational behaviour. Instead, users are forced to postulate equality proofs and appeal to them explicitly to simplify expressions, making axioms dramatically more complicated to work with than built-in primitives. On the other hand, the equality reflection rule from extensional type theory solves these problems by collapsing computation and equality, at the cost of having no practical type checking algorithm. This paper introduces Rewriting Type Theory (RTT), a type theory where it is possible to add computational assumptions in the form of rewrite rules. Rewrite rules go beyond the computational capabilities of intensional type theory, but in contrast to extensional type theory, they are applied automatically so type checking does not require input from the user. To ensure type soundness of RTT—as well as effective type checking—we provide a framework where confluence of user-defined rewrite rules can be checked modularly and automatically, and where adding new rewrite rules is guaranteed to preserve subject reduction. The properties of RTT have been formally verified using the MetaCoq framework and an implementation of rewrite rules is already available in the Agda proof assistant.

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Practical Subtyping for Curry-Style Languages

Lepigre

Raffalli

2019

ACM Trans. Program. Lang. Syst.

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We present a new, syntax-directed framework for Curry-style type systems with subtyping. It supports a rich set of features, and allows for a reasonably simple theory and implementation. The system we consider has sum and product types, universal and existential quantifiers, and inductive and coinductive types. The latter two may carry size invariants that can be used to establish 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. One of the key ideas is to 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. Termination is then obtained using a standard (semantic) normalisation proof. To demonstrate the practicality of the system, we provide an implementation accepting all the examples discussed in the article.

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Size-based termination of higher-order rewriting

Cited by 4 publications

References 149 publications

Dependency Pairs Termination in Dependent Type Theory Modulo Rewriting

Dependency Pairs Termination in Dependent Type Theory Modulo Rewriting

The taming of the rew: a type theory with computational assumptions

Practical Subtyping for Curry-Style Languages

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