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Proof Assistant Decision Procedures for Formalizing Origami [chapter]

Cezary Kaliszyk, Tetsuo Ida
2011 Lecture Notes in Computer Science  
We examine the strength of Gröbner Bases implementations comparing proof assistants with each other and with the implementations provided in computer algebra systems.  ...  Proofs performed with computer algebra systems can be accompanied by proof documents, still they lack complete mathematical rigorousity, like the one provided by proof assistant checked proofs.  ...  Computer Algebra functionality in Proof Assistants Mainstream Computer Algebra Systems, for instance Mathematica and Maple, are weakly founded.  ... 
doi:10.1007/978-3-642-22673-1_4 fatcat:n3dfqj73afgphagwrc3umn4yhi

A short proof of the Dimension Conjecture for real hypersurfaces in C^2 [article]

Alexander Isaev, Boris Kruglikov
2015 arXiv   pre-print
Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in C^2 via the dimension of the algebra of infinitesimal automorphisms.  ...  In this note, we propose a short argument for obtaining their result.  ...  Beloshapka for communicating to us an example of a hypersurface in C 2 with dim hol(M, p) = 1, for which he kindly performed Maple-assisted computations.  ... 
arXiv:1509.09139v1 fatcat:gfsqdqojcna7xkjfptu2ggorae

Developments in Formal Proofs [article]

Thomas C. Hales
2014 arXiv   pre-print
Finally, we discuss advances in the automation of formal proofs, as implemented in proof assistants such as Mizar, Coq, Isabelle, and HOL Light.  ...  This report describes three particular technological advances in formal proofs. The HOL Light proof assistant will be used to illustrate the design of a highly reliable system.  ...  I would also like to thank the many speakers and participants at the special program on Semantics of proofs and certified math at IHP.  ... 
arXiv:1408.6474v1 fatcat:ho7xhssjurcidhy4yfxphnqbm4

Reasoning About Algebraic Structures with Implicit Carriers in Isabelle/HOL [chapter]

Walter Guttmann
2020 Lecture Notes in Computer Science  
We prove Chen and Grätzer's construction theorem for Stone algebras in Isabelle/HOL.  ...  The development requires extensive reasoning about algebraic structures in addition to reasoning in algebraic structures.  ...  In this paper we focus on Isabelle/HOL, which is the higherorder logic instance of the generic proof assistant Isabelle [29] .  ... 
doi:10.1007/978-3-030-51054-1_14 fatcat:7oikixg6rrdhdeiwl3j7k64fty

Mathematics in the Age of the Turing Machine [article]

Thomas Hales
2013 arXiv   pre-print
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.  ...  The structures of abstract algebra -groups, rings, modules, algebras, algebraically closed fields and so forth -have all been laid out formally in the Coq proof assistant.  ...  Another praiseworthy project is Kaliszyk and Wiedijk's implementation of a computer algebra system on top of the proof assistant HOL Light.  ... 
arXiv:1302.2898v1 fatcat:ko2iqnfnq5dlbe6fwirbpvuswy

Interactive Theorem Proving

Gerwin Klein, Ruben Gamboa
2016 Journal of automated reasoning  
This special issue collects current efforts towards the construction of formal proofs with the use of interactive theorem provers, which combine formal proof-checking and prooffinding tools with human  ...  The paper describes how abstract interpretation can be used to bridge this gap automatically, and how this process has been verified in the Coq proof assistant.  ...  The paper by Bourke, van Glabbeek, and Höfner, "Mechanizing a Process Algebra for Network Protocols," describes an approach to modeling Mobile Ad hoc Networks and Wireless Mesh Networks in Isabelle/HOL  ... 
doi:10.1007/s10817-016-9363-7 fatcat:cz5nkkk6hfbathwh4rg5s5urum

Formalising Mathematics – in Praxis; A Mathematician's First Experiences with Isabelle/HOL and the Why and How of Getting Started

Angeliki Koutsoukou-Argyraki
2020 Jahresbericht der Deutschen Mathematiker-Vereinigung (Teubner)  
and potential of proof assistants for mathematics.  ...  This is an account of a mathematician's first experiences with the proof assistant (interactive theorem prover) Isabelle/HOL, including a discussion on the rationale behind formalising mathematics and  ...  The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.  ... 
doi:10.1365/s13291-020-00221-1 fatcat:pt4sfc3hlvfwlant3b4v2varwu

Formalisation of Selected Results from Group Theory

Joseph Johannes Thommes, Manuel Eberl, Tobias Nipkow
2021 Zenodo  
This thesis deals with the formalisation of some group-theoretic results in Isabelle/HOL – an interactive theorem prover that machine-checks every proof step.  ...  – the canonical Isabelle/HOL library on abstract algebra.  ...  The Isabelle proof assistant "Isabelle is a generic proof assistant.  ... 
doi:10.5281/zenodo.4785363 fatcat:ggo4zojijrgk5kfuufsxxsfsgu

Towards computer aided mathematics

Christoph Benzmüller
2006 Journal of Applied Logic  
, and computer algebra systems, into interactive proof assistants.  ...  The aim of this project is to reconstruct, formalize, and fully verify Hales complete proof in the assistance system HOL-LIGHT.  ... 
doi:10.1016/j.jal.2005.10.001 fatcat:gawws4hbbnappewt2z4dao5loq

Tool-Based Verification of a Relational Vertex Coloring Program [chapter]

Rudolf Berghammer, Peter Höfner, Insa Stucke
2015 Lecture Notes in Computer Science  
: the automated theorem prover Prover9 and the two proof assistants Coq and Isabelle/HOL.  ...  We present different approaches of using a special purpose computer algebra system and theorem provers in software verification.  ...  Verification of Proof Obligations using Isabelle/HOL In this section we discuss the verification of the relational program (VC) by means of Isabelle/HOL, a proof assistant that additionally offers support  ... 
doi:10.1007/978-3-319-24704-5_17 fatcat:v25bbisshfe5feqimxf2u5ocm4


2011 Mathematical Structures in Computer Science  
in variants of predicate calculus.  ...  The best we could do is relative consistency proofs, such as Gödel's result that the continuum hypothesis is indeed consistent with ZFC.  ...  The full and successful development of this piece of mathematics in the HOL Light proof assistant is thus an important landmark in formal mathematics.  ... 
doi:10.1017/s0960129511000235 fatcat:f5ehf7oyjjdrffke7wt3lpujc4

Proving Bounds for Real Linear Programs in Isabelle/HOL [chapter]

Steven Obua
2005 Lecture Notes in Computer Science  
For our work described in this paper, we have chosen the generic proof assistant Isabelle, tailored to Higher-Order Logic (HOL) [4] .  ...  In the following, we will refer to this environment as Isabelle/HOL. An important step in Hales' proof is the maximization of about 10 5 real linear programs.  ...  Using a rewriting oracle that is based on the ideas found in [7] , these operations can be performed so quickly that it is projected that the linear programs arising in the proof of the Kepler conjecture  ... 
doi:10.1007/11541868_15 fatcat:7v4m2yzgxnb7deoqie5sqmadqm

Proof Certificates for Algebra and Their Application to Automatic Geometry Theorem Proving [chapter]

Benjamin Grégoire, Loïc Pottier, Laurent Théry
2011 Lecture Notes in Computer Science  
Integrating decision procedures in proof assistants in a safe way is a major challenge.  ...  The method is implemented in the Coq system but, since our specialised version of Buchberger's algorithm outputs explicit proof certificates, it could be easily adapted to other proof assistants.  ...  We have compared our method with two systems: Macaulay2 [10], a system dedicated to algebraic geometry which is very efficient in Gröbner bases computations and HOL Light [14] , a proof assistant that  ... 
doi:10.1007/978-3-642-21046-4_3 fatcat:ssvv2zgqena3joe3tdtr7sus6y

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.  ...  We give an overview of our formalizations in the proof assistant Isabelle/HOL of certain irrationality and transcendence criteria for infinite series from three different research papers: by Erdős and  ...  The proof of this celebrated theorem is long and elaborate; it has not been formalized in any proof assistant, to our knowledge.  ... 
arXiv:2101.05257v2 fatcat:ozo6fkwhc5aujaxynmy4a2ztjy

A Methodology for the Formal Verification of Dynamic Fault Trees using HOL Theorem Proving

Yassmeen Elderhalli, Osman Hasan, Sofiene Tahar
2019 IEEE Access  
In this paper, we further enrich the DFT library in HOL by providing the formalization of spare gates with a shared spare and the verification details of their probabilistic behavior.  ...  Leveraging upon the expressive and sound nature of higher-order logic (HOL) theorem proving, it has been recently proposed for the analysis of DFTs algebraically.  ...  The proof script for the drive-by-wire system is around 4950 lines long. B. FORMAL VERIFICATION OF CARDIAC ASSIST SYSTEM The DFT for the cardiac assist system (CAS) is shown in Fig. 8 [20] .  ... 
doi:10.1109/access.2019.2942829 fatcat:27c4t2ss7rawhhj5so2x6d6w6a
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