A Simplified Proof of the Church–Rosser Theorem

Komori, Yuichi; Matsuda, Naosuke; Yamakawa, Fumika

doi:10.1007/s11225-013-9470-y

Cited by 8 publications

(5 citation statements)

References 3 publications

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“…Hence, if there exists an arbitrary reduction strategy * that satisfies both properties, then the main lemma can be established. In fact, the main lemma holds even for β η-equality, because for β η-reduction, under an inside-out development we still have Lemma 5, Proposition 4, and Proposition 2 without bounds as observed already in [11]. This implies that under a general framework with such a strategy, it is possible to analyze quantitative properties of rewriting systems in the exactly same way, and indeed λ -calculus with β η-reduction and weakly orthogonal higher-order rewriting systems [17,5] are instances of these systems.…”

Section: Concluding Remarks and Further Worksupporting

confidence: 63%

“…To overcome this point, an improved version of Theorem 2 is introduced such that * -translation is applied only when new redexes are indeed reduced. The basic idea of the second method V-size is essentially the same as the proof given in [11]. In summary, the functions bl and CR-red including a common reduct are respectively defined by induction on the length of one side of the peak, and V-size is by induction on that of both sides of the peak.…”

Section: Quantitative Analysis Of Church-rosser For β -Reductionmentioning

confidence: 99%

See 1 more Smart Citation

On Upper Bounds on the Church-Rosser Theorem

Fujita

2017

Electron. Proc. Theor. Comput. Sci.

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The Church-Rosser theorem in the type-free lambda-calculus is well investigated both for beta-equality and beta-reduction. We provide a new proof of the theorem for beta-equality with no use of parallel reductions, but simply with Takahashi's translation (Gross-Knuth strategy). Based on this, upper bounds for reduction sequences on the theorem are obtained as the fourth level of the Grzegorczyk hierarchy.Comment: In Proceedings WPTE 2016, arXiv:1701.0023

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Section: Concluding Remarks and Further Worksupporting

confidence: 63%

Section: Quantitative Analysis Of Church-rosser For β -Reductionmentioning

confidence: 99%

On Upper Bounds on the Church-Rosser Theorem

Fujita

2017

Electron. Proc. Theor. Comput. Sci.

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“…The proof is relatively standard. We use the Takahashi translation technique due to Komori et al (2014), which is a simplification of the standard parallel reduction technique. It uses the Takahashi translation e † of terms e, defined as the simultaneous single-step reduction of all βζ δ ιμν-redexes of e in left-most inner-most order.…”

Section: Theorem 31 (Confluence) If G mentioning

confidence: 99%

Is sized typing for Coq practical?

Chan¹,

LI²,

Bowman³

2023

J. Funct. Prog.

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Contemporary proof assistants such as Coq require that recursive functions be terminating and corecursive functions be productive to maintain logical consistency of their type theories, and some ensure these properties using syntactic checks. However, being syntactic, they are inherently delicate and restrictive, preventing users from easily writing obviously terminating or productive functions at their whim. Meanwhile, there exist many sized type theories that perform type-based termination and productivity checking, including theories based on the Calculus of (Co)Inductive Constructions (CIC), the core calculus underlying Coq. These theories are more robust and compositional in comparison. So why haven’t they been adapted to Coq? In this paper, we venture to answer this question with CIC $\widehat{\ast}$ , a sized type theory based on CIC. It extends past work on sized types in CIC with additional Coq features such as global and local definitions. We also present a corresponding size inference algorithm and implement it within Coq’s kernel; for maximal backward compatibility with existing Coq developments, it requires no additional annotations from the user. In our evaluation of the implementation, we find a severe performance degradation when compiling parts of the Coq standard library, inherent to the algorithm itself. We conclude that if we wish to maintain backward compatibility, using size inference as a replacement for syntactic checking is impractical in terms of performance.

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“…One of the simplest applications of the compositional Z is for the βη-reduction on the untyped lambda calculus (although it can be directly proved by the original Z theorem, as Komori et al (2013) showed). If we let → 1 be → η , → 2 be → β , f 1 be the complete development of → η , and f 2 be the complete development of → β , then these satisfy the conditions of the compositional Z theorem.…”

Section: A Mapping F Satisfies the Z Property For → If And Only If It...mentioning

confidence: 99%

“…If we define a → f (a). The Z theorem has been applied to some variants of the lambda calculus in Dehornoy and van Oostrom (2008), Komori et al (2013), Accattoli and Kesner (2012), Nakazawa and Nagai (2014).…”

Section: Introductionmentioning

confidence: 99%