Executable Matrix Operations on Matrices of Arbitrary Dimensions

We provide the operations of matrix addition, multiplication, transposition, and matrix comparisons as executable functions over ordered semirings. Moreover, it is proven that strongly normalizing (monotone) orders can be lifted to strongly normalizing (monotone) orders over matrices. We further show that the standard semirings over the naturals, integers, and rationals, as well as the arctic semirings satisfy the axioms that are required by our matrix theory. Our formalization was performed as part of the IsaFoR/CeTA-system [3] which contains several termination techniques. The provided theories have been essential to formalize matrix-interpretations [1] and arctic interpretations [2]. A short description of this formalization can be found in [4].