I´m just thinking aloud, but, because incorporating deeper mathematics concepts has proven to be the best solution for better and more flexible programming languages with fewer errors, I wonder if raising the type classes incorporating axioms can solve additional problems. At first sight it does: class Abelian a where (+) :: a -> a -> a property ((+))= a+b == b+a this permits: 1- safer polimorphism: I can safely reuse the operator + if the type and the property is obeyed. The lack of ability to redefine operators is a problem for DSLs that must use wreid symbols combinations with unknow meanings. To use common operators with fixed properties is very good. the same aplies for method names. 2- the compiler can use the axions as rewrite rules. 3- in debugging mode, it is possible to verify the axiom for each value a generated during execution. Thus, a generator is not needed like in quickcheck. The logic to quickcheck can be incorporated in the debugging executable. 3 guaranties that 1 and 2 are safe. a type class can express a relation between types, but it is not possible to define relation between relations.