Validating Mathematical Structures

Sharing of notations and theories across an inheritance hierarchy of mathematical structures, e.g., groups and rings, is important for productivity when formalizing mathematics in proof assistants. The packed classes methodology is a generic design pattern to define and combine mathematical structures in a dependent type theory with records. When combined with mechanisms for implicit coercions and unification hints, packed classes enable automated structure inference and subtyping in hierarchies, e.g., that a ring can be used in place of a group. However, large hierarchies based on packed classes are challenging to implement and maintain. We identify two hierarchy invariants that ensure modularity of reasoning and predictability of inference with packed classes, and propose algorithms to check these invariants. We implement our algorithms as tools for the Coq proof assistant, and show that they significantly improve the development process of Mathematical Components, a library for formalized mathematics.