ECC, an extended calculus of constructions

A higher-order calculus ECC (extended calculus of constructions) is presented which can be seen as an extension of the calculus of constructions by adding strong sum types and a fully cumulative type hierarchy. ECC turns out to be rather expressive so that mathematical theories can be abstractly described and abstract mathematics may be adequately formalized. It is shown that ECC is strongly normalizing and has other nice proof-theoretic properties. An ω-set (realizability) model is described to show how the essential properties of the calculus can be captured set-theoretically