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Foundational extensible corecursion: a proof assistant perspective

Jasmin Christian Blanchette, Andrei Popescu, Dmitriy Traytel
2015 Proceedings of the 20th ACM SIGPLAN International Conference on Functional Programming - ICFP 2015  
The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool.  ...  This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity.  ...  In contrast to Agda and Coq, proof assistants based on higherorder logic (HOL), such as HOL4, HOL Light, and Isabelle/HOL, generally adhere to the foundational approach.  ... 
doi:10.1145/2784731.2784732 dblp:conf/icfp/Blanchette0T15 fatcat:hkvtk6jlirc5tfube3qro34gdm

Foundational extensible corecursion: a proof assistant perspective

Jasmin Christian Blanchette, Andrei Popescu, Dmitriy Traytel
2015 SIGPLAN notices  
The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool.  ...  This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity.  ...  In contrast to Agda and Coq, proof assistants based on higherorder logic (HOL), such as HOL4, HOL Light, and Isabelle/HOL, generally adhere to the foundational approach.  ... 
doi:10.1145/2858949.2784732 fatcat:iplmxzsh7nhbdo6qlmopn5dsky

Foundational Extensible Corecursion [article]

Jasmin Christian Blanchette and Andrei Popescu and Dmitriy Traytel
2015 arXiv   pre-print
The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool.  ...  This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity.  ...  proof assistant.  ... 
arXiv:1501.05425v1 fatcat:nhtd7stvcrgrvpwnwl5yzmxi4i

Corecursion and Non-divergence in Session-Typed Processes [chapter]

Bernardo Toninho, Luis Caires, Frank Pfenning
2014 Lecture Notes in Computer Science  
Our approach builds on a Curry-Howard correspondence between our type system and linear logic extended with co-inductive types, for which our non-divergence property implies consistency.  ...  In this paper we address, by means of typing, the challenge of ensuring non-divergence in a session-typed π-calculus with general (co)recursion, while still allowing interesting infinite behaviors to be  ...  Concluding Remarks We have developed a theory of coinductive definitions for a synchronous π-calculus with corecursion, establishing a logical foundation based on intuitionistic linear logic which provides  ... 
doi:10.1007/978-3-662-45917-1_11 fatcat:yfkdtylwzbc5zcfmolkpragg7y

Formal Languages, Formally and Coinductively [article]

Dmitriy Traytel
2017 arXiv   pre-print
In this article, we study this representation in the Isabelle proof assistant. We define regular operations on infinite tries and prove the axioms of Kleene algebra for those operations.  ...  Thereby, we exercise corecursion and coinduction and confirm the coinductive view being profitable in formalizations, as it improves over the set-of-words view with respect to proof automation.  ...  Non-Primitive Corecursion in Proof Assistants. Automation for corecursion in proof assistants is much less developed than its recursive counterpart.  ... 
arXiv:1611.09633v2 fatcat:r5wup2poajd53dd2i4vw7oftem

Corecursion and Non-Divergence in Session Types

Bernardo Toninho, Luis Caires, Frank Pfenning
2018
Our approach builds on a Curry-Howard correspondence between our type system and linear logic extended with co-inductive types, for which our non-divergence property implies consistency.  ...  In this paper we address, by means of typing, the challenge of ensuring non-divergence in a session-typed π-calculus with general (co)recursion, while still allowing interesting infinite behaviors to be  ...  Concluding Remarks We have developed a theory of coinductive definitions for a synchronous π-calculus with corecursion, establishing a logical foundation based on intuitionistic linear logic which provides  ... 
doi:10.1184/r1/6604511 fatcat:lo3yoaqrlna37ei2e62u6bpxea

A Coinductive Animation of Turing Machines [chapter]

Alberto Ciaffaglione
2015 Lecture Notes in Computer Science  
We adopt corecursion and coinduction to formalize Turing Machines and their operational semantics in the proof assistant Coq.  ...  Our effort may be seen as a first step towards the formal development of basic computability theory. 1 As said above, we deal with deterministic TMs, i.e., non-ambiguous specifications: for every pair  ...  In fact, a bisimilarity proof is just an infinite proof term built by corecursion (hence, it must respect the same guardedness constraint that corecursive functions have to).  ... 
doi:10.1007/978-3-319-15075-8_6 fatcat:oxh5bnngijddzfqmdaikjjwq24

Coinduction Plain and Simple [article]

François Bry
2020 arXiv   pre-print
The article finally suggests in its conclusion extensions of functional and logic programming with limited and decidable forms of the generalized coinduction proof principle.  ...  Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata.  ...  Acknowledgments The author thanks Schloss Dagstuhl -Leibniz Rechenzentrum für Informatik and Alexander Artikis, Thomas Eiter, Alessandro Margara, and Stijn Vansummeren, the organisers of the Dagstuhl Seminar "Foundations  ... 
arXiv:2007.09909v2 fatcat:l5euzvldcrgmdl2wpbtrll7kpq

Partiality and recursion in interactive theorem provers – an overview

ANA BOVE, ALEXANDER KRAUSS, MATTHIEU SOZEAU
2014 Mathematical Structures in Computer Science  
Other systems and logics are covered to a certain extent, but not exhaustively.  ...  The use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalizing mathematics is becoming more common and feasible in practice.  ...  proof assistants.  ... 
doi:10.1017/s0960129514000115 fatcat:y3gdgilgd5bvlev3haw4l3fyxi

Universality of Proofs (Dagstuhl Seminar 16421)

Gilles Dowek, Catherine Dubois, Brigitte Pientka, Florian Rabe, Marc Herbstritt
2017 Dagstuhl Reports  
The seminar was motivated by the fact that it is nowadays difficult to exchange proofs from one proof assistant to another one.  ...  Thus a formal proof cannot be considered as a universal proof, reusable in different contexts.  ...  We show how the Nelson-Oppen method is a particular case of our MCSat-combination method, reconciling the widely-implemented technique with the new MCSat ideas.  ... 
doi:10.4230/dagrep.6.10.75 dblp:journals/dagstuhl-reports/DowekDPR16 fatcat:5isaoju43faxfdxb3tx7xoi7py

Wellfounded recursion with copatterns

Andreas M. Abel, Brigitte Pientka
2013 SIGPLAN notices  
assistants such as Coq and Agda.  ...  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  ...  we have introduced copatterns as a novel perspective on defining infinite structures that might serve as a new foundation for coinduction in dependentlytyped languages, overcoming the problems in the  ... 
doi:10.1145/2544174.2500591 fatcat:ixomcdxzunfd7gajbi3aj7pmhq

Wellfounded recursion with copatterns

Andreas M. Abel, Brigitte Pientka
2013 Proceedings of the 18th ACM SIGPLAN international conference on Functional programming - ICFP '13  
assistants such as Coq and Agda.  ...  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  ...  we have introduced copatterns as a novel perspective on defining infinite structures that might serve as a new foundation for coinduction in dependentlytyped languages, overcoming the problems in the  ... 
doi:10.1145/2500365.2500591 dblp:conf/icfp/AbelP13 fatcat:4nifaqt3irbizkeg2hnlhdzp74

Well-founded recursion with copatterns and sized types

ANDREAS ABEL, BRIGITTE PIENTKA
2016 Journal of functional programming  
data in proof assistants such as Coq and Agda.  ...  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.  ...  Big thanks to the reviewers of a previous version of this article who asked many insightful questions that lead to the present, reworked version. The first author  ... 
doi:10.1017/s0956796816000022 fatcat:4j25e64w45crbf6euk3pnmnpfm

Generalizing Inference Systems by Coaxioms [chapter]

Davide Ancona, Francesco Dagnino, Elena Zucca
2017 Lecture Notes in Computer Science  
Besides axioms and inference rules with the usual meaning, a generalized inference system allows coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree.  ...  A B S T R A C T After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types.  ...  proof assistants, such as Coq and Agda.  ... 
doi:10.1007/978-3-662-54434-1_2 fatcat:ei6k3q5ncnfofncahwbdshh5om

Generalizing inference systems by coaxioms [article]

Francesco Dagnino
2018 arXiv   pre-print
Besides axioms and inference rules with the usual meaning, a generalized inference system allows coaxioms, which are, intuitively, axioms which can only be applied "at infinite depth" in a proof tree.  ...  After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types.  ...  proof assistants, such as Coq and Agda.  ... 
arXiv:1712.01014v2 fatcat:5pwyr5qmmrhhdasp6prjb5dbjm
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