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Algebraic Numbers in Isabelle/HOL [chapter]

René Thiemann, Akihisa Yamada
2016 Lecture Notes in Computer Science  
Based on existing libraries for matrices, factorization of integer polynomials, and Sturm's theorem, we formalized algebraic numbers in Isabelle/HOL.  ...  Moreover, we provide two implementations to display algebraic numbers, an injective one that reveals the representing polynomial, or an approximative one that only displays a fixed amount of digits.  ...  The absence of an equivalent to Sturm's method for the complex numbers in Isabelle/HOL prevented us from having native support for complex algebraic numbers.  ... 
doi:10.1007/978-3-319-43144-4_24 fatcat:3c4l45go2behrl7b2gbhkt4p44

A Verified Implementation of Algebraic Numbers in Isabelle/HOL

Sebastiaan J. C. Joosten, René Thiemann, Akihisa Yamada
2018 Journal of automated reasoning  
We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers.  ...  We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s10817-018-09504-w pmid:32226180 pmcid:PMC7089722 fatcat:pvzb5tg36jdb5bfrvnlgmajq7y

Algebraically Closed Fields in Isabelle/HOL [chapter]

Paulo Emílio de Vilhena, Lawrence C. Paulson
2020 Lecture Notes in Computer Science  
We fill this gap by documenting its formalisation in Isabelle/HOL, describing the difficulties that impeded this development and their solutions.  ...  A fundamental theorem states that every field admits an algebraically closed extension. Despite its central importance, this theorem has never before been formalised in a proof assistant.  ...  Anthony Bordg for his invaluable guidance, and we also thank Martin Baillon who was directly involved in the formalisation effort and who was always available to having fruitful discussions, as a colleague  ... 
doi:10.1007/978-3-030-51054-1_12 fatcat:2ysccy5du5eafoawj6nkwrufzy

Algebras for Program Correctness in Isabelle/HOL [chapter]

Alasdair Armstrong, Victor B. F. Gomes, Georg Struth
2014 Lecture Notes in Computer Science  
We present a reference formalisation of Kleene algebra and demonic refinement algebra with tests in Isabelle/HOL. It provides three different formalisations of tests.  ...  It includes an algebraic account of Hoare logic for partial correctness and several refinement and concurrency control laws in a total correctness setting.  ...  Demonic Refinement Algebra in Isabelle We now sketch our formalisation of demonic refinement algebra in the theorem proving environment Isabelle/HOL [18] .  ... 
doi:10.1007/978-3-319-06251-8_4 fatcat:jmwp5mqm5beyhekkirkn7oybuu

Data Refinement in Isabelle/HOL [chapter]

Florian Haftmann, Alexander Krauss, Ondřej Kunčar, Tobias Nipkow
2013 Lecture Notes in Computer Science  
The paper shows how the code generator of Isabelle/HOL supports data refinement, i.e., providing efficient code for operations on abstract types, e.g., sets or numbers.  ...  In order to automate the process of setting up specific data refinements, two packages for transferring definitions and theorems between types are exploited.  ...  Code Generation Isabelle/HOL supports code generation for a number of functional programming languages (SML, OCaml, Haskell, Scala).  ... 
doi:10.1007/978-3-642-39634-2_10 fatcat:x3hpfvxuarbytciwnhipxiso2m

Cardinals in Isabelle/HOL [chapter]

Jasmin Christian Blanchette, Andrei Popescu, Dmitriy Traytel
2014 Lecture Notes in Computer Science  
We report on a formalization of ordinals and cardinals in Isabelle/HOL.  ...  We also discuss several applications of this general theory in formal developments.  ...  Our formalization is performed in Isabelle/HOL [12] , an implementation enriched with Haskell-style type classes [6] .  ... 
doi:10.1007/978-3-319-08970-6_8 fatcat:iwslhjcasrakda4cz4dtfstsjm

From LCF to Isabelle/HOL

Lawrence C. Paulson, Tobias Nipkow, Makarius Wenzel
2019 Formal Aspects of Computing  
It is generic, supporting a number of different formal calculi, but by far the most important of these is its instantiation to higher-order logic: Isabelle/HOL.  ...  Here, we focus on Isabelle/HOL and its distinctive strengths.  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s00165-019-00492-1 fatcat:rv5zmo22fjedvjxjz4lm22v6de

Generating Verified LLVM from Isabelle/HOL

Peter Lammich, Michael Wagner
2019 International Conference on Interactive Theorem Proving  
We present a framework to generate verified LLVM programs from Isabelle/HOL.  ...  It is based on a code generator that generates LLVM text from a simplified fragment of LLVM, shallowly embedded into Isabelle/HOL.  ...  Moreover, an array access still required a significant amount of overhead compared to a simple pointer dereference in C. 22:2 Generating Verified LLVM from Isabelle/HOL The next step towards more efficient  ... 
doi:10.4230/lipics.itp.2019.22 dblp:conf/itp/Lammich19 fatcat:mska6m2pavc6jf2tl2hlumi2du

Three Chapters of Measure Theory in Isabelle/HOL [chapter]

Johannes Hölzl, Armin Heller
2011 Lecture Notes in Computer Science  
We lift this restriction by introducing the extended real numbers. We define the Borel σ-algebra for an arbitrary type forming a topological space.  ...  We define the Lebesgue measure using the gauge integral available in Isabelle's multivariate analysis.  ...  Paulson for porting Coble's measure space formalization from HOL4 to Isabelle/HOL and to Robert Himmelmann for formalizing the first iteration of the Lebesgue measure.  ... 
doi:10.1007/978-3-642-22863-6_12 fatcat:2ab2xe6ahnhw3b5qpim3ixn6ym

Deciding Univariate Polynomial Problems Using Untrusted Certificates in Isabelle/HOL

Wenda Li, Grant Olney Passmore, Lawrence C. Paulson
2017 Journal of automated reasoning  
We present a proof procedure for univariate real polynomial problems in Isabelle/HOL. The core mathematics of our procedure is based on univariate cylindrical algebraic decomposition.  ...  We present experiments demonstrating the efficacy of this approach, in many cases yielding orders of magnitude improvements over previous methods.  ...  distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in  ... 
doi:10.1007/s10817-017-9424-6 fatcat:hmwbpdzomvf4vgctwnerib4tp4

Unifying Theories in Isabelle/HOL [chapter]

Abderrahmane Feliachi, Marie-Claude Gaudel, Burkhart Wolff
2010 Lecture Notes in Computer Science  
In this paper, we present various extensions of Isabelle/HOL by theories that are essential for several formal methods.  ...  First, we explain how we have developed an Isabelle/HOL theory for a part of the Unifying Theories of Programming (UTP). It contains the theories of alphabetized relations and designs.  ...  of the specification and theorem proving environment Isabelle/HOL.  ... 
doi:10.1007/978-3-642-16690-7_9 fatcat:cnuqqi62jjeu3k6hubbfmp47jy

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

Predicate Transformer Semantics for Hybrid Systems: Verification Components for Isabelle/HOL [article]

Jonathan Julián Huerta y Munive, Georg Struth
2021 arXiv   pre-print
We present a semantic framework for the deductive verification of hybrid systems with Isabelle/HOL.  ...  It supports reasoning about the temporal evolutions of hybrid programs in the style of differential dynamic logic modelled by flows or invariant sets for vector fields.  ...  /HOL for fruitful discussions.  ... 
arXiv:1909.05618v3 fatcat:7lnrnamltvcutgxfezop4akt7q

Bounded Model Generation for Isabelle/HOL

Tjark Weber
2005 Electronical Notes in Theoretical Computer Science  
The algorithm has been implemented in the interactive theorem prover Isabelle/HOL, where it is used to automatically generate countermodels for non-theorems.  ...  Isabelle/HOL runs on a number of different platforms, and installation should be as simple as possible. Therefore we have also implemented a naive DPLL-based [5, 21] SAT solver in Isabelle.  ...  This allows for the automatic generation of finite countermodels for non-theorems in Isabelle/HOL.  ... 
doi:10.1016/j.entcs.2004.10.027 fatcat:pg34wda2kvgb7co7i67hry2qgy

A Verified Implementation of B+-Trees in Isabelle/HOL [article]

Niels Mündler, Tobias Nipkow
2022 arXiv   pre-print
In this paper we present the verification of an imperative implementation of the ubiquitous B+-tree data structure in the interactive theorem prover Isabelle/HOL.  ...  The imperative implementation is verified in two steps: an abstract set interface is refined to an executable but inefficient purely functional implementation which is further refined to the efficient  ...  It may thus find applications in the development of libraries where a verified implementation of a set interface is needed.  ... 
arXiv:2208.09066v1 fatcat:ozvtcbqllbgkhmxrwnnb7btpce
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