A simply typed context calculus with rst-class environments

We introduce a simply typed -calculus " which has both contexts and environments as rst-class values. In ", holes in contexts are represented by ordinary variables of appropriate types and hole lling is represented by the functional application together with a new abstraction mechanism which takes care of packing and unpacking of the term which is used to ll in the holes of the context. " is a conservative extension of the simply typed -calculus, enjoys subject reduction property, is con uent and strongly normalizing. The traditional method of de ning substitution does not work for our calculus. So, we also introduce a new method of de ning substitution. Although we introduce the new de nition of substitution out of necessity, the new de nition turns out to be conceptually simpler than the traditional de nition of substitution.