A constructive algebraic hierarchy in Coq
Journal of Symbolic Computation - Integrated reasoning and algebra systems
Interactive Theorem Proving and Program Development
Interactive Theorem Proving and Program Development
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Journal of Automated Reasoning
Towards Constructive Homological Algebra in Type Theory
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Formal Aspects of Computing
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ICFEM '09 Proceedings of the 11th International Conference on Formal Engineering Methods: Formal Methods and Software Engineering
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Formal Aspects of Computing
A modular formalisation of finite group theory
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
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Mathematical Structures in Computer Science
Developing the algebraic hierarchy with type classes in coq
ITP'10 Proceedings of the First international conference on Interactive Theorem Proving
Incidence simplicial matrices formalized in Coq/SSReflect
MKM'11 Proceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics
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EUROCAST'11 Proceedings of the 13th international conference on Computer Aided Systems Theory - Volume Part I
Verifying an algorithm computing discrete vector fields for digital imaging
CICM'12 Proceedings of the 11th international conference on Intelligent Computer Mathematics
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Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
Computing persistent homology within Coq/SSReflect
ACM Transactions on Computational Logic (TOCL)
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In this paper, we present a complete formalization in the Coq theorem prover of an important algorithm in computational algebra, namely the calculation of the effective homology of a bicomplex. As a necessary tool, we encode a hierarchy of algebraic structures in constructive type theory, including graded and infinite data structures. The experience shows how some limitations of the Coq proof assistant to deal with this kind of algebraic data can be overcome by applying a separation of concerns principle; more concretely, we propose to distinguish in the representation of an algebraic structure (such as a group or a module) a behavioural part, containing operation signatures and axioms, and a structural part determining if the algebraic data is free, of finite type and so on.