Inductive and Coinductive Components of Corecursive Functions in Coq

Bertot, Yves; Komendantskaya, Ekaterina

doi:10.1016/j.entcs.2008.05.018

Cited by 17 publications

(19 citation statements)

References 33 publications

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“…AmiCo implements the first general, provably sound, and fully automatic method for mixing recursive and corecursive calls in function definitions. The idea of mixing recursion and corecursion appears in Bertot [11] for the stream filter, and a generalization is sketched in Bertot and Komendantskaya [13] for corecursion up to constructors. Leino's Dafny tool [46] was the first to offer such a mixture for general codatatypes, which turned out to be unsound and was subsequently restricted to the sound but limited fragment of tail recursion.…”

Section: Related Work and Discussionmentioning

confidence: 99%

Friends with Benefits

Blanchette

Bouzy

Lochbihler

et al. 2017

Programming Languages and Systems

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Abstract. We introduce AmiCo, a tool that extends a proof assistant, Isabelle/ HOL, with flexible function definitions well beyond primitive corecursion. All definitions are certified by the assistant's inference kernel to guard against inconsistencies. A central notion is that of friends: functions that preserve the productivity of their arguments and that are allowed in corecursive call contexts. As new friends are registered, corecursion benefits by becoming more expressive. We describe this process and its implementation, from the user's specification to the synthesis of a higher-order definition to the registration of a friend. We show some substantial case studies where our approach makes a difference.

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Section: Related Work and Discussionmentioning

confidence: 99%

Friends with Benefits

Blanchette

Bouzy

Lochbihler

et al. 2017

Programming Languages and Systems

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“…A more subtle analysis of the corecursive functions that fail to satisfy the guardedness condition * can be found in [14,4,21,7]. In particular, the mentioned papers offer a solution to the problem of formalising productive corecursive functions of this kind.…”

Section: Position Is Guarded If It Occurs As a Direct Sub-term Of A Cmentioning

confidence: 99%

“…Therefore the value of this paper, as well as (e.g.) [5,7,8] is in the technical elegance and practical implementation in the existing proof assistants.…”

Section: Structurally Recursive Functionsmentioning

confidence: 99%

“…Efforts have been made to extend the basic guarded corecursion in the same spirit that well-founded recursion extends the basic structural recursion. We can mention [14] and [4,7], which basically incorporate well-founded recursion to make sure several non-productive recursive calls are allowed as long as they ultimately become productive. In particular, [14] introduces a generalization of the concept of well-founded relation that uses an extra dimension to cover at the same time recursive or co-recursive functions; since there is an extra dimension, two notions of limits can be used and recursive values can mix terminating recursive and productive co-recursive aspects in a seamless fashion.…”

Section: Introductionmentioning

confidence: 99%

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Using Structural Recursion for Corecursion

Bertot

Komendantskaya

2009

Lecture Notes in Computer Science

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Abstract. We propose a (limited) solution to the problem of constructing stream values defined by recursive equations that do not respect the guardedness condition. The guardedness condition is imposed on definitions of corecursive functions in Coq, AGDA, and other higher-order proof assistants. In this paper, we concentrate in particular on those nonguarded equations where recursive calls appear under functions. We use a correspondence between streams and functions over natural numbers to show that some classes of non-guarded definitions can be modelled through the encoding as structural recursive functions. In practice, this work extends the class of stream values that can be defined in a constructive type theory-based theorem prover with inductive and coinductive types, structural recursion and guarded corecursion.

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“…We start with the following program: We can then create the transition system demonstrating type-correctness and reify to obtain the following functions: As we can see here, the always, eventually notion of productivity that is given in [3] is recast as a coinductive function meeting the guardedness condition, with auxiliary inductive functions representing the eventually criterion.…”

Section: Examplementioning

confidence: 99%

Cyclic Proofs and Coinductive Principles

Mendel-Gleason¹,

Hamilton²

EPiC Series in Computing

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It is possible to provide a proof for a coinductive type using a corecursive function coupled with a guardedness condition. The guardedness condition, however, is quite restrictive and many programs which are in fact productive and do not compromise soundness will be rejected. We present a system of cyclic proof for an extension of System F extended with sums, products and (co)inductive types. Using program transformation techniques we are able to take some programs whose productivity is suspected and transform them, using a suitable theory of equivalence, into programs for which guardedness is syntactically apparent. The equivalence of the proof prior and subsequent to transformation is given by a bisimulation relation.

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Inductive and Coinductive Components of Corecursive Functions in Coq

Cited by 17 publications

References 33 publications

Friends with Benefits

Friends with Benefits

Using Structural Recursion for Corecursion

Cyclic Proofs and Coinductive Principles

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