Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: mtrl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: plasm-ph