Model Driven Engineering (MDE) uses models to represent the core part of the software systems. The Unified Model Language (UML) is a widely accepted standard for modeling software systems. Although UML provides numbers of concepts and diagrams to describe the system, there is still an unsolved problem that the semantics and refinement relations of models are not formally defined. In this paper, we apply the constructive type theory to formalize the class diagrams and object diagrams. A suitable subset of UML static models is identified and formally defined. The theorem assistant Coq is applied to encode the semantics of class diagrams. Moreover the refinement relations are also formalized in Coq. The whole approach is supported by tools that do not constrain the semantic definition's expressiveness and flexibility while making it machine-checkable. Our approach offers a novel way for giving a precise foundation in UML and contributes to the goal of improving the overall trustworthy software systems by combining theoretical and practical techniques.