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Formalizing O Notation in Isabelle/HOL [chapter]

Jeremy Avigad, Kevin Donnelly
2004 Lecture Notes in Computer Science  
We describe a formalization of asymptotic O notation using the Isabelle/HOL proof assistant.  ...  Our goal here is to describe an implementation of one important asymptotic notion -"big O notation" -using the Isabelle/HOL proof assistant.  ...  Moreover, in Isabelle/HOL types are grouped together into sorts. These sorts are ordered, and types of a sort inherit properties of any of the sort's ancestors.  ... 
doi:10.1007/978-3-540-25984-8_27 fatcat:47xtoamb25g4thhtj4c2a4a67y

Irrationality and Transcendence Criteria for Infinite Series in Isabelle/HOL [article]

Angeliki Koutsoukou-Argyraki, Wenda Li, Lawrence C. Paulson
2021 arXiv   pre-print
Our formalizations in Isabelle/HOL can be found on the Archive of Formal Proofs.  ...  Here we describe selected aspects of the formalization and discuss what this reveals about the use and potential of Isabelle/HOL in formalizing modern mathematical research, particularly in these parts  ...  preliminaries were already available in Isabelle/HOL.  ... 
arXiv:2101.05257v2 fatcat:ozo6fkwhc5aujaxynmy4a2ztjy

Verifying a Solver for Linear Mixed Integer Arithmetic in Isabelle/HOL [chapter]

Ralph Bottesch, Max W. Haslbeck, Alban Reynaud, René Thiemann
2020 Lecture Notes in Computer Science  
We implement a decision procedure for linear mixed integer arithmetic and formally verify its soundness in Isabelle/HOL.  ...  The procedure itself is then implemented as a branch-and-bound algorithm, and is available in several languages via Isabelle's code generator.  ...  The authors are listed in alphabetical order regardless of individual contributions or seniority.  ... 
doi:10.1007/978-3-030-55754-6_14 fatcat:e5mgdyqrczbv5hkv6aulm2tyru

Formalizing O Notation in Isabelle / HOL

Jeremy Avigad, Kevin Donnelly
2018
We describe a formalization of asymptotic O notation using the Isabelle / HOL proof assistant.  ...  Our goal here is to describe an implementation of one important asymptotic notion -"big O notation" -using the Isabelle/HOL proof assistant.  ...  Moreover, in Isabelle/HOL types are grouped together into sorts. These sorts are ordered, and types of a sort inherit properties of any of the sort's ancestors.  ... 
doi:10.1184/r1/6491615 fatcat:lgb2nxq7y5filez5xuqbjwfysa

Verification Framework for Control System Functionality of Unmanned Aerial Vehicles [article]

Omar A. Jasim, Sandor M. Veres
2020 arXiv   pre-print
To illustrate the proceedures, a higher-order logic interactive theorem-prover and an automated theorem-prover are utilized to formally verify a nonlinear attitude control system of a generic multi-rotor  ...  using Isabelle/HOL prover.  ...  Isabelle/HOL expressions and symbols are described in Table 1 .  ... 
arXiv:2006.10860v1 fatcat:uc2ecw5dg5bn3hznxjsslimsqi

On the Formalisation of $$\varSigma $$ -Protocols and Commitment Schemes [chapter]

David Butler, David Aspinall, Adrià Gascón
2019 Research Series on the Chinese Dream and China's Development Path  
Every definition and every theorem presented in this paper has been checked mechanically by the Isabelle/HOL proof assistant.  ...  In this work we provide the first formal analysis in a proof assistant of such a relationship and in doing so formalise Σ-protocols and commitment schemes and provide proofs of security for well known  ...  We formalise this in Isabelle/HOL, but the structure could be used in other proof assistants.  ... 
doi:10.1007/978-3-030-17138-4_8 dblp:conf/post/ButlerAG19 fatcat:razflr2bbnd65amqwt5x3xfqqu

Coquelicot: A User-Friendly Library of Real Analysis for Coq

Sylvie Boldo, Catherine Lelay, Guillaume Melquiond
2014 Mathematics in Computer Science  
As such, its support is warranted in proof assistants, so that the users have a way to formally verify mathematical theorems and correctness of critical systems.  ...  The Coq system comes with an axiomatization of standard real numbers and a library of theorems on real analysis. Unfortunately, this standard library is lacking some widely used results.  ...  Acknowledgements The authors are grateful to Pierre Michalak andÉvelyne Roudneff for allowing us to take the Baccalaureate exam in real-life conditions in a high school in Massy, and for organizing the  ... 
doi:10.1007/s11786-014-0181-1 fatcat:eptv7v543bg63nlm6gzyakcctu

A Perron–Frobenius Theorem for Deciding Matrix Growth

René Thiemann
2021 Journal of Logical and Algebraic Methods in Programming  
We further verify both the algorithm and the new Perron-Frobenius theorem in the proof assistant Isabelle/HOL, and integrate it into Ce T A, a verified certifier for various properties, including complexity  ...  Because of the new results, Ce T A no longer requires a verified implementation of algebraic numbers.  ...  We thank Manuel Eberl for his hints on performing reasoning on real asymptotics within Isabelle. Finally, we thank the anonymous reviewers for their helpful remarks.  ... 
doi:10.1016/j.jlamp.2021.100699 fatcat:tfarrkr4xzb6jozn74yisrpmbi

A Formal Proof of the Irrationality of ζ(3) [article]

Assia Mahboubi, Thomas Sibut-Pinote
2021 arXiv   pre-print
The rest of the proof combines arithmetical ingredients and asymptotic analysis, which we conduct by extending the Mathematical Components libraries.  ...  We formally prove this result by an a posteriori verification of calculations performed by computer algebra algorithms in a Maple session.  ...  We also thank Cyril Cohen, Pierre Roux and Enrico Tassi for their help, in particular with the libraries this work depends on.  ... 
arXiv:1912.06611v6 fatcat:kijni3uvvjcpjhvm4esnq3bcgy

Directed Security Policies: A Stateful Network Implementation

Cornelius Diekmann, Lars Hupel, Georg Carle
2014 Electronic Proceedings in Theoretical Computer Science  
The solution scales to large networks, which is confirmed by a large real-world case study. Its correctness is guaranteed by the Isabelle/HOL theorem prover.  ...  In particular, we provide a criterion to verify the lack of side effects in linear time.  ...  The Isabelle/HOL theory files can be obtained at https://github.com/diekmann/topoS.  ... 
doi:10.4204/eptcs.150.3 fatcat:a2clel3vlrhjlosyqvc2othpsa

Formalising $$\varSigma $$-Protocols and Commitment Schemes Using CryptHOL

D. Butler, A. Lochbihler, D. Aspinall, A. Gascón
2020 Journal of automated reasoning  
We formalise this proof at an abstract level using the modularity available in Isabelle/HOL and CryptHOL. This way, the proofs of the instantiations come for free.  ...  $$\varSigma $$ Σ -protocols allow a prover to convince a verifier that they possess some knowledge without leaking information about the knowledge.  ...  Isabelle/HOL Isabelle/HOL is an interactive theorem prover that implements Higher Order Logic (HOL).  ... 
doi:10.1007/s10817-020-09581-w fatcat:plg6seffhvf5xat2e5mwssxamm

Efficient certification of complexity proofs: formalizing the Perron–Frobenius theorem (invited talk paper)

Jose Divasón, Sebastiaan Joosten, Ondřej Kunčar, René Thiemann, Akihisa Yamada
2018 Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018  
To cover the theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of theorems between both libraries.  ...  In this work we formalize the Perron-Frobenius theorem.  ...  The authors are listed in alphabetical order regardless of individual contributions or seniority. We thank Fabian Immler for his explanations on how to perform continuity proofs in the HMA library.  ... 
doi:10.1145/3176245.3167103 fatcat:dbeagv66k5gttinzjqhw6xnzbu

Efficient certification of complexity proofs: formalizing the Perron–Frobenius theorem (invited talk paper)

Jose Divasón, Sebastiaan Joosten, Ondřej Kunčar, René Thiemann, Akihisa Yamada
2018 Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018  
To cover the theorem in its full extent, we establish a connection between two different Isabelle/HOL libraries on matrices, enabling an easy exchange of theorems between both libraries.  ...  In this work we formalize the Perron-Frobenius theorem.  ...  The authors are listed in alphabetical order regardless of individual contributions or seniority. We thank Fabian Immler for his explanations on how to perform continuity proofs in the HMA library.  ... 
doi:10.1145/3167103 dblp:conf/cpp/DivasonJKT018 fatcat:niku3pjyhvdhfchc6p2ikanrua

Formal Asymptotic Analysis of Online Scheduling Algorithms for Plug-In Electric Vehicles' Charging

Asad Ahmed, Osman Hasan, Falah Awwad, Nabil Bastaki, Syed Rafay Hasan
2018 Energies  
Generally, the computational complexity of an algorithm is computed using asymptotic analysis.  ...  A large-scale integration of plug-in electric vehicles (PEVs) into the power grid system has necessitated the design of online scheduling algorithms to accommodate the after-effects of this new type of  ...  The O-notation has been formalized in the Isabelle/HOL (higher-order-logic) theorem prover [22] using the ring theory.  ... 
doi:10.3390/en12010019 fatcat:dfzezxhsujau5ns7ak2q7arvni

Formalizing Monotone Algebras for Certification of Termination and Complexity Proofs [chapter]

Christian Sternagel, René Thiemann
2014 Lecture Notes in Computer Science  
To be able to certify these proofs, we formalized several kinds of interpretations in the proof assistant Isabelle/HOL.  ...  We report on our integration of matrix interpretations, arctic interpretations, and nonlinear polynomial interpretations over various domains, including the reals.  ...  The authors are listed in alphabetical order regardless of individual contributions or seniority.  ... 
doi:10.1007/978-3-319-08918-8_30 fatcat:g4ujbxaodzdyjcgti33xum5t44
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