An Intuitionistic Set-theoretical Model of the Extended Calculus of Constructions

Werner’s set-theoretical model is one of the most intuitive models of ECC. It combines a functional view of predicative universes with a collapsed view of the impredicative sort Prop. However this model of Prop is so coarse that the principle of excluded middle holds. In this paper, we interpret Prop into a topological space (a special case of Heyting algebra) to make it more intuitionistic without sacrificing simplicity. We prove soundness and show some applications of our model.