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## 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.