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- research-articleJuly 2014
A Certified Reduction Strategy for Homological Image Processing
ACM Transactions on Computational Logic (TOCL), Volume 15, Issue 3Article No.: 23, Pages 1–23https://doi.org/10.1145/2630789The analysis of digital images using homological procedures is an outstanding topic in the area of Computational Algebraic Topology. In this article, we describe a certified reduction strategy to deal with digital images, but one preserving their ...
- articleJune 2014
Formalization of Shannon's Theorems
Journal of Automated Reasoning (JAUR), Volume 53, Issue 1Pages 63–103https://doi.org/10.1007/s10817-013-9298-1The most fundamental results of information theory are Shannon's theorems. These theorems express the bounds for (1) reliable data compression and (2) data transmission over a noisy channel. Their proofs are non-trivial but are rarely detailed, even in ...
- research-articleNovember 2013
Computing persistent homology within Coq/SSReflect
ACM Transactions on Computational Logic (TOCL), Volume 14, Issue 4Article No.: 26, Pages 1–16https://doi.org/10.1145/2528929Persistent homology is one of the most active branches of computational algebraic topology with applications in several contexts such as optical character recognition or analysis of point cloud data. In this article, we report on the formal development ...
- ArticleJuly 2013
ML4PG in computer algebra verification
CICM'13: Proceedings of the 2013 international conference on Intelligent Computer MathematicsPages 354–358https://doi.org/10.1007/978-3-642-39320-4_28ML4PG is a machine-learning extension that provides statistical proof hints during the process of Coq/SSReflect proof development. In this paper, we use ML4PG to find proof patterns in the CoqEAL library - a library that was devised to verify the ...
- ArticleDecember 2012
Coherent and strongly discrete rings in type theory
CPP'12: Proceedings of the Second international conference on Certified Programs and ProofsPages 273–288https://doi.org/10.1007/978-3-642-35308-6_21We present a formalization of coherent and strongly discrete rings in type theory. This is a fundamental structure in constructive algebra that represents rings in which it is possible to solve linear systems of equations. These structures have been ...
- ArticleJuly 2012
Verifying an algorithm computing discrete vector fields for digital imaging
CICM'12: Proceedings of the 11th international conference on Intelligent Computer MathematicsPages 216–230https://doi.org/10.1007/978-3-642-31374-5_15In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological ...
- ArticleMay 2012
Towards a certified computation of homology groups for digital images
CTIC'12: Proceedings of the 4th international conference on Computational Topology in Image ContextPages 49–57https://doi.org/10.1007/978-3-642-30238-1_6In this paper we report on a project to obtain a verified computation of homology groups of digital images. The methodology is based on programming and executing inside the Coq proof assistant. Though more research is needed to integrate and make ...
- ArticleAugust 2011
Point-free, set-free concrete linear algebra
ITP'11: Proceedings of the Second international conference on Interactive theorem provingPages 103–118Abstract linear algebra lets us reason and compute with collections rather than individual vectors, for example by considering entire subspaces. Its classical presentation involves a menagerie of different settheoretic objects (spaces, families, ...
- ArticleJuly 2011
A Formal Semantics of PLC Programs in Coq
COMPSAC '11: Proceedings of the 2011 IEEE 35th Annual Computer Software and Applications ConferencePages 118–127https://doi.org/10.1109/COMPSAC.2011.23Programmable logic Controllers (PLC) are embedded systems that are widely used in industry. We propose a formal semantics of the Instruction List (IL) language, one of the five programing languages defined in the IEC 61131-3 standard for PLC programing. ...
- ArticleJuly 2010
Formally verified conditions for regularity of interval matrices
We propose a formal study of interval analysis that concentrates on theoretical aspects rather than on computational ones. In particular we are interested in conditions for regularity of interval matrices. An interval matrix is called regular if all ...
- ArticleAugust 2009
Packaging Mathematical Structures
TPHOLs '09: Proceedings of the 22nd International Conference on Theorem Proving in Higher Order LogicsPages 327–342https://doi.org/10.1007/978-3-642-03359-9_23This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the <Emphasis Type="SmallCaps">Coq</Emphasis> system. This alternative to telescopes in particular ...
- ArticleJuly 2009
Finite Groups Representation Theory with Coq
Calculemus '09/MKM '09: Proceedings of the 16th Symposium, 8th International Conference. Held as Part of CICM '09 on Intelligent Computer MathematicsPages 438–452https://doi.org/10.1007/978-3-642-02614-0_34Representation theory is a branch of algebra that allows the study of groups through linear applications, i.e. matrices. Thus problems in abstract groups can be reduced to problems on matrices. Representation theory is the basis of character theory. In ...