Section: New Results

Formal proofs on linear algebra

Participants : Guillaume Cano, Maxime Dénès, Anders Mörtberg [University of Chalmers, Sweden] , Vincent Siles [University of Chalmers, Sweden] , Yves Bertot.

This year we completed a work on matrix canonical forms, providing formal proofs for the following results:

• Smith normal forms of matrices on principal ideal domains are unique,

• Every matrix on a field is similar to its Frobenius normal form

• Every matrix on an algebraically closed field is similar to its Jordan normal form

We also studied techniques to combine high-level mathematical descriptions and proofs of algorithms with executable implementations. This work led to a publication at ITP'12 [10] . We are still working on extending this work to rational numbers and real algebraic numbers.

We then worked on tools to automate proofs. In the ring tactic, all elements considered must belong to the same type. We worked on extending this tactic to dependent families of types, like the type of matrices where each dimension gives rise to a different type in the family and multiplications typically concern matrices of different types, while remaining associative.