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The greatest common divisor: A case study for program extraction from classical proofs
[chapter]

1996
*
Lecture Notes in Computer Science
*

Here we treat that example as

doi:10.1007/3-540-61780-9_60
fatcat:ohln4bijzrbzhndjx6gfg5vbaq
*a**case**study**for**program**extraction**from**classical**proofs*. We apply H. ...*The*number c ∈ (*a*1 ,*a*2 ) dividing*a*1 and*a*2 is*the**greatest**common**divisor*since any*common**divisor*of*a*1 and*a*2 must also be*a**divisor*of c. ... We would like to thank Monika Seisenberger and Felix Joachimski*for*implementing*the*refined*A*-translation and doing*the*gcd example in*the*MINLOG system. ...##
###
Greatest common divisors of polynomials given by straight-line programs

1988
*
Journal of the ACM
*

Both

doi:10.1145/42267.45069
fatcat:hjux5gwgybe5nikgp3nozu7zzq
*the*algorithm*for*that construction and*the**greatest**common**divisor*algorithm are in random polynomial time*for**the*usual coefftcient fields and output*a*straight-line*program*, which with controllably ...*The*second result shows how to find*a*straight-line*program**for**the*reduced numerator and denominator*from*one*for**the*corresponding rational function. ... Introduction This*study*is concerned with complexity questions about performing operations, such as*greatest**common**divisor*(GCD) computation and factorization, on multivariate polynomials. ...##
###
Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges

2018
*
Proceedings of the London Mathematical Society
*

Our

doi:10.1112/plms.12181
fatcat:z3m527zkojexrlywsfov5ccxza
*proof*uses*the*circle method and some oscillatory integral estimates (following*a*paper of Zhan) to reduce matters to establishing some mean-value estimates*for*certain Dirichlet polynomials associated ...*The*Type d_4 sum is treated similarly using*the**classical*L^2 mean value theorem and*the**classical*van der Corput exponential sum estimates. ... We use (*a*, b) and [*a*, b]*for**the**greatest**common**divisor*and least*common*multiple of natural numbers*a*, b respectively, and write*a*|b if*a*divides b. ...##
###
Page 7528 of Mathematical Reviews Vol. , Issue 98M
[page]

1998
*
Mathematical Reviews
*

*The*

*greatest*

*common*

*divisor*:

*a*

*case*

*study*

*for*

*program*

*extraction*

*from*

*classical*

*proofs*. Types

*for*

*proofs*and

*programs*(Torino, 1995), 36-46, Lecture Notes in Comput. ... Here we treat that example as

*a*

*case*

*study*

*for*

*program*

*extraction*

*from*

*classical*

*proofs*. We apply H. Friedman’s

*A*-translation [in Higher set theory (Proc. Conf., Math. ...

##
###
Page 2070 of Mathematical Reviews Vol. , Issue 98D
[page]

1998
*
Mathematical Reviews
*

Schwichtenberg,

*The**greatest**common**divisor*:*a**case**study**for**program**extraction**from**classical**proofs*(36-46); Ilya Beylin and Peter Dybjer,*Extracting**a**proof*of coherence*for*monoidal cat- egories*from*...*a**proof*of normalization*for*monoids (47-61); Jan Cederquist and Sara Negri,*A*constructive*proof*of*the*Heine- Borel covering theorem*for*formal reals (62-75); Thierry Coquand and Jan M. ...##
###
Euclid's Number-Theoretical Work
[article]

2010
*
arXiv
*
pre-print

Therefore, Euclid has preliminarily established Theory of Divisibility and

arXiv:0902.2465v2
fatcat:k2xehcvr4vdclpwvc332nhheqa
*the**greatest**common**divisor*. This is*the*foundation of Number Theory. ... We were quite surprised when we began to read*the*Elements*for**the*first time. ... Acknowledgements I am happy to record my gratitude to my supervisor Professor Xiaoyun Wang who made*a*lot of helpful suggestions. I wish to thank*the*referees*for**a*careful reading of this paper. ...##
###
Foundations of Garside Theory
[article]

2013
*
arXiv
*
pre-print

This text consists of

arXiv:1309.0796v1
fatcat:iw46d4gqjnd4hpft55h7qw2tbq
*the*introduction, table of contents, and bibliography of*a*long manuscript (703 pages) that is currently submitted*for*publication. ... This manuscript develops an extension of Garside's approach to braid groups and provides*a*unified treatment*for**the*various algebraic structures that appear in this context. ...*the**divisors*of*the*fundamental element ∆ n :*the*point is that, if g is an element of B + n , then there exists*a*(unique)*greatest**common**divisor*g 1*for*g and ∆ n and, moreover g = 1 implies g 1 = 1 ...##
###
The Logic of Computer Programming

1978
*
IEEE Transactions on Software Engineering
*

In our writing we have drawn on

doi:10.1109/tse.1978.231499
fatcat:ylzqmhqefjfyrnacqqozolyffi
*the*wealth of material about*the**greatest**common**divisor*and ... We are also grateful to*the*following members of*the*Theory of Computing Panel of*the*NSF Computer Science and Engineering Research*Study*(COSERS) --Richard Karp,*A*lbert Meyer, John Reynolds, Robert Ritchie ... As*a*new specimen*for*our*study*of recursion we will Introduce*a*recursive cousin of*the**greatest**common**divisor*algorithm of Euclid, which appeared in his Elements over 2200 years ago. ...##
###
A Formally Certified End-to-End Implementation of Shor's Factorization Algorithm
[article]

2022
*
arXiv
*
pre-print

Techniques familiar to most programmers

arXiv:2204.07112v1
fatcat:7guc72okn5gkboopsiop4zn2l4
*from**the**classical*domain*for*avoiding, discovering, and diagnosing bugs do not easily transfer, at scale, to*the*quantum domain because of its unique characteristics ...*The*proof's validity is automatically confirmed -- certified -- by*a*"*proof*assistant". ... Acknowledgement We thank Andrew Childs, Steven Girvin, Liang Jiang, and Peter Shor*for*helpful feedback on*the*manuscript. ...##
###
Computers and Thought Lecture: The Ubiquity of Discovery

1977
*
International Joint Conference on Artificial Intelligence
*

One of them («2) says to look

dblp:conf/ijcai/Lenat77
fatcat:dtqksvygdngqddw3ghtzyd6wkq
*for*motivation at*the*extreme*cases*of various known relationships.*The*relation in this*case*is "*Divisors*of". ... , new pieces of mass spectroscopy theory.These rules are then usable by*the*Dendral*program*, as if they had been*extracted**from**a*human expert.>*The*PECOS*program*[Barstow 1977] contains rules about computer ...##
###
A Framework for the Verification of Certifying Computations

2013
*
Journal of automated reasoning
*

*A*checker

*for*such

*a*witness is usually much simpler than

*the*original algorithmyet it is all

*the*user has to trust. ... ., formal

*proofs*that outputs

*for*particular inputs are correct. We do so by combining

*the*concept of certifying algorithms with methods

*for*code verification and theorem proving. ... Moreover, we thank Lars Noschinski

*for*developing

*a*powerful graph library in Isabelle/HOL. ...

##
###
A Framework for the Verification of Certifying Computations
[article]

2013
*
arXiv
*
pre-print

*A*checker

*for*such

*a*witness is usually much simpler than

*the*original algorithm - yet it is all

*the*user has to trust. ... We use

*the*automatic verifier VCC

*for*establishing

*the*correctness of

*the*checker and

*the*interactive theorem prover Isabelle/HOL

*for*high-level mathematical properties of algorithms. ... Moreover, we thank Lars Noschinski

*for*developing

*a*powerful graph library in Isabelle/HOL. ...

##
###
Homology of gaussian groups

2003
*
Annales de l'Institut Fourier
*

x of M admits

doi:10.5802/aif.1951
fatcat:rtfbjgxfknfktheaqcnv6boog4
*a*unique decomposition x = xl ~ ~ ~ xp such that,*for*each i, xi is*the**greatest*left*divisor*of xi ... xp lying in X. ... However,*the*rightmost cell does not start*from*1, but*from*as shown in Figure 6 .*Proof*.*The**case*n = 0 is obvious, so assume n > 1. We use induction on*the*length of*the*word w. ...##
###
Program development in constructive type theory

1992
*
Theoretical Computer Science
*

We present

doi:10.1016/0304-3975(92)90037-g
fatcat:cwdzm2e265awdpbribnyfeodey
*the**program*development concept in*a*logical framework including constructive type theory and then show how to use such theories to derive*programs**from**proofs*of formal specifications. ... We are interested in two important facts that are*the*mechanization of*the**proof*construction and*the*possibility to express in*the*theory significant concepts*for**programming*(like inductively defined ... Example 2 Another example is to determine*the**greatest**common**divisor*of two nonzero naturals. ...##
###
QArith: Coq Formalisation of Lazy Rational Arithmetic
[chapter]

2004
*
Lecture Notes in Computer Science
*

This formalisation highlights

doi:10.1007/978-3-540-24849-1_20
fatcat:i4uwc4sggnbq7duac57togtyqq
*the*rôle of type theory both as*a*tool to verify hand-written*programs*and as*a*tool to generate verified*programs*. ... Here*the*function abs is*the*forgetful projection*from*Z onto nat (Coq natural numbers). Thus*for*p and q two integers, makeQ p q produces*the*signed binary sequence corresponding to p q . ... This work was completed during*the*first author's visit to INRIA Sophia Antipolis, made possible by*a*grant*from**the*Dutch Organization*for*Scientific Research (NWO). ...
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