An extended type system with lambda-typed lambda-expressions (extended version)

We present the type system $\mathtt{d}$, an extended type system with lambda-typed lambda-expressions. It is related to type systems originating from the Automath project. $\mathtt{d}$ extends existing lambda-typed systems by an existential abstraction operator as well as propositional operators. $\beta$-reduction is extended to also normalize negated expressions using a subset of the laws of classical negation, hence $\mathtt{d}$ is normalizing both proofs and formulas which are handled uniformly as functional expressions. $\mathtt{d}$ is using a reflexive typing axiom for a constant $\tau$ to which no function can be typed. Some properties are shown including confluence, subject reduction, uniqueness of types, strong normalization, and consistency. We illustrate how, when using $\mathtt{d}$, due to its limited logical strength additional axioms must be added both for negation and for the mathematical structures whose deductions are to be formalized. Several appendices deal with extensions and variations of the proposed system.