Commutative algebra in the Mizar system
Journal of Symbolic Computation - Special issue on computer algebra and mechanized reasoning: selected St. Andrews' ISSAC/Calculemus 2000 contributions
Journal of Functional Programming
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Packaging Mathematical Structures
TPHOLs '09 Proceedings of the 22nd International Conference on Theorem Proving in Higher Order Logics
A modular formalisation of finite group theory
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
A HOL theory of euclidean space
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
Proving bounds for real linear programs in Isabelle/HOL
TPHOLs'05 Proceedings of the 18th international conference on Theorem Proving in Higher Order Logics
How to make ad hoc proof automation less ad hoc
Proceedings of the 16th ACM SIGPLAN international conference on Functional programming
PVS linear algebra libraries for verification of control software algorithms in C/ACSL
NFM'12 Proceedings of the 4th international conference on NASA Formal Methods
Coherent and strongly discrete rings in type theory
CPP'12 Proceedings of the Second international conference on Certified Programs and Proofs
Computing persistent homology within Coq/SSReflect
ACM Transactions on Computational Logic (TOCL)
Canonical structures for the working coq user
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
A machine-checked proof of the odd order theorem
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
The rooster and the butterflies
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
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Abstract 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, mappings), whose use often involves tedious and non-constructive pointwise reasoning; this is in stark contrast with the regularity and effectiveness of the matrix computations hiding beneath abstract linear algebra. In this paper we show how a simple variant of Gaussian elimination can be used to model abstract linear algebra directly, using matrices only to represent all categories of objects, with operations such as subspace intersection and sum. We can even provide effective support for direct sums and subalgebras. We have formalized this work in Coq, and used it to develop all of the group representation theory required for the proof of the Odd Order Theorem, including results such as the Jacobson Density Theorem, Clifford's Theorem, the Jordan-Holder Theorem for modules, theWedderburn Structure Theorem for semisimple rings (the basis for character theory).