## Section: New Results

### First-class simultaneous substitutions in the two-level logic approach

Participant : Kaustuv Chaudhuri.

The *two-level logic approach* that underlies the Abella prover
is excellent at reasoning about the inductive structure of terms with
binding constructs, such as $\lambda $-terms from the
$\lambda $-calculus. However, there is no built in support in Abella
for reasoning about the inductive structure of (simultaneous)
substitutions. This lack of this kind of support is often criticized
in the $\lambda $-tree syntax representational style that is used in
Abella; indeed, in a number of other systems based on this style,
support for reasoning about substitutions is explicitly added into the
trusted kernel. In [14] we show how to formalize
substitutions in Abella in a fluent and high level manner, where all
the meta-theory can be proven in a straightforward manner. We
illustrate its use in giving a clean formulation of fact that the Howe
extension of applicative similarity is a pre-congruence, a standard
result from the meta-theory of the $\lambda $-calculus that requires
sophistication in treating simultaneous substitutions.