HOL4 is the latest version of the HOL interactive proof assistant for higher order logic: a programming environment in which theorems can be proved and proof tools implemented. Built-in decision procedures and theorem provers can automatically establish many simple theorems (users may have to prove the hard theorems themselves!) An oracle mechanism gives access to external programs such as SAT and BDD engines. HOL 4 is particularly suitable as a platform for implementing combinations of deduction, execution and property checking. several widely used versions of the HOL system: 1. HOL88 from Cambridge; 2. HOL90 from Calgary and Bell Labs; 3. HOL98 from Cambridge, Glasgow and Utah. HOL 4 is the successor to these. Its development was partly supported by the PROSPER project. HOL 4 is based on HOL98 and incorporates ideas and tools from HOL Light. The ProofPower system is another implementation of HOL.
HOL Light is a computer program to help users prove interesting mathematical theorems completely formally in higher order logic. It sets a very exacting standard of correctness, but provides a number of automated tools and pre-proved mathematical theorems (e.g. about arithmetic, basic set theory and real analysis) to save the user work. It is also fully programmable, so users can extend it with new theorems and inference rules without compromising its soundness. There are a number of versions of HOL, going back to Mike Gordon's work in the early 80s. Compared with other HOL systems, HOL Light uses a much simpler logical core and has little legacy code, giving the system a simple and uncluttered feel. Despite its simplicity, it offers theorem proving power comparable to, and in some areas greater than, other versions of HOL, and has been used for some significant industrial-scale verification applications.
* Higher-Order Logic * o HOL (Higher-Order Logic) o HOLCF (Higher-Order Logic of Computable Functions) * First-Order Logic * o FOL (Many-sorted First-Order Logic) o ZF (Set Theory) o CCL (Classical Computational Logic) o LCF (Logic of Computable Functions) o FOLP (FOL with Proof Terms) * Miscellaneous * o Sequents (first-order, modal and linear logics) o CTT (Constructive Type Theory) o Cube (The Lambda Cube)
LambdaCLAM is a tool for automated theorem proving in higher order domains. In particular LambdaCLAM specialises in proof using induction based on the rippling heuristic. LambdaCLAM is a higher-order version of CLAM. Both CLAM and LambdaCLAM use proof planning to guide the search for a proof A proof plan is a proof of a theorem at some level of abstraction presented as a tree. Each node in this tree is justified by a tactic. The exact nature of these tactics is unspecified, they may be sequences of inference rules, programs for generating sequences of inferences or a further proof plan at some lower level of abstraction. In principle while the generation of the proof tree may have involved heuristics and (possibly) unsound inference steps, it can be justified by executing the tactics attached to the nodes.
Vampire is winning at least one division of the world cup in theorem proving CASC since 1999. All together Vampire won 17 titles: more than any other prover. We traditionally take part in the following two divisions of the competition: * The FOF division: unrestricted first-order problems. This division was ranked second in importance after the MIX division before 2007 and is now recognised as the main competition division. * The CNF division: first-order problems in conjunctive normal form. This division was called MIX before 2007 and recognised as the main competition division. We also participate in other, more special competition divisions but Vampire is not specialised for them so our achievements are mostly modest.
In my experience, proof readers tend to be rather calm individuals, going about their work in an unruffled, dignified manner. Proof readers are rarely confrontational in temperament, because proofreading by its very nature requires a serene and reflective approach. So, it was rare for me, as an Operations Manager supervising, amongst other people, proof readers, to have to intervene in any kind of serious dispute.
Except when it came to hyphens.
A Special Issue on Formal Proof
Using computers in proofs both extends mathematics with new results and creates new mathematical questions about the nature and technique of such proofs. This special issue features a collection of articles by practitioners and theorists of such formal proofs which explore both aspects.
(pp. 1363)
Thomas Hales
(pp. 1370)
Formal Proof--The Four-Color Theorem
Georges Gonthier
(pp. 1382)
Formal Proof--Theory and Practice
John Harrison
(pp. 1395)
Formal Proof--Getting Started
Freek Wiedijk
By Julie Rehmeyer
Web edition : Friday, November 14th, 2008
Mathematicians develop computer proof-checking systems in order to realize century-old dreams of fully precise, accurate mathematics.
ProofWeb is both a system for teaching logic and for using proof assistants through the web.
ProofWeb can be used in three ways. First, one can use the guest login, for which one does not even need to register. Secondly, a user can be a student in a logic or proof assistants course. We are hosting courses free of charge. If you are a teacher and would like to host your course on this server, send email to proofweb@cs.ru.nl. Thirdly, if teachers do not want to trust us with their students' files, they can freely download the ProofWeb system and run it on a server of their own.
K. Heng. (2014)cite arxiv:1404.6248Comment: Published in American Scientist: Volume 102, Number 3, Pages 174 to 177 (http://www.americanscientist.org/issues/pub/2014/3/the-nature-of-scientific-proof-in-the-age-of-simulations).
H. Cao, and X. Zhu. (2006)cite arxiv:math/0612069Comment: This is a revised version of the article by the same authors that originally appeared in Asian J. Math., 10(2) (2006), 165--492.
E. Stark. Foundations of Software Technology and Theoretical Computer Science, volume 206 of Lecture Notes in Computer Science, Springer Berlin / Heidelberg, (1985)
J. Søgaard-Andersen, S. Garl, J. Guttag, N. Lynch, and A. Pogosyants. PROCEEDINGS OF THE 5TH INTERNATIONAL CONFERENCE ON COMPUTER AIDED VERIFICATION, ELOUNDA, GREECE, VOLUME 697 OF LECTURE, page 305--319. Springer Verlag, (1993)