A Higher-Order Calculus and Theory Abstraction

We present a higher-order calculus ECC which naturally combines Coquand-Huet's calculus of constructions and Martin-LL of's type theory with universes. ECC is very expressive, both for structured abstract reasoning and for program speciication and construction. In particular , the strong sum types together with the type universes provide a useful module mechanism for abstract description of mathematical theories and adequate formalization of abstract mathematics. This allows comprehensive structuring of interactive development of speciications, programs and proofs. After a summary of the meta-theoretic properties of the calculus, an !?Set (realizability) model of ECC is described to show how its essential properties can be captured set-theoretically. The model construction entails the logical consistency of the calculus and gives some hints on how to adequately formalize abstract mathematics. Theory abstraction in ECC is discussed as a pragmatic application.