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UML static models are expressed using a mixture of class diagrams and OCL expressions. In a well formed static model, the OCL expressions and class diagrams are type consistent. Checking for type consistency of static models involves both inclusion and parametric polymorphism. This paper defines a semantics of type consistency in terms of a type theory for UML static models. The type theory is shown to be correct with respect to a value semantics for OCL. The existence of a consistency checking algorithm for UML static models is established.