Validating Mathematical Structures.

With regard to formalizing mathematics in proof assistants, the hierarchy of mathematical structure that allows for the sharing of notations and theories, and makes subtyping of structures implicit, is a key ingredient of infrastructure to support the users. The packed classes method is a generic design pattern to define and combine mathematical structures in a dependent type theory with records. The Coq proof assistant has mechanisms to enable automated structure inference and subtyping with packed classes; that is, implicit coercions and canonical structures. In this paper, we identify the invariants of hierarchies that ensure the modularity of reasoning and the predictability of inference with packed classes, propose checking tools for those invariants, and show that our tools significantly improve the development process of Mathematical Components, a formalized mathematics library for Coq.