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

Abstract: 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.
 Comments: 150 pages Subjects: Logic in Computer Science (cs.LO) Cite as: arXiv:1803.06488 [cs.LO] (or arXiv:1803.06488v1 [cs.LO] for this version)

## Submission history

From: Matthias Weber [view email]
[v1] Sat, 17 Mar 2018 11:01:23 GMT (91kb)