We present the 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 type 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.Comment: for extended version, see arXiv:1803.06488

@article{abs-1803-10143,
  title = {An extended type system with lambda-typed lambda-expressions},
  author = {Matthias Weber},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 16, Issue 4},
  year = {2020},
  doi = {10.23638/lmcs-16(4:12)2020}
}