## Section: New Results

### Mechanized metatheory revisited

Participant : Dale Miller.

When proof assistants and theorem provers implement the metatheory of
logical systems, they must deal with a range of syntactic expressions
(e.g., types, formulas, and proofs) that involve variable bindings.
Since most mature proof assistants do not have built-in methods to
treat bindings, they have been extended with various packages and
libraries that allow them to encode such syntax using, for example, De
Bruijn numerals. In the paper, [10], Miller
puts forward the argument that bindings are such an intimate aspect of
the structure of expressions that they should be accounted for
directly in the underlying programming language support for proof
assistants and not via packages and libraries. He presents an
approach to designing programming languages and proof assistants that
directly supports bindings in syntax. The roots of this approach can
be found in the *mobility* of binders between term-level
bindings, formula-level bindings (quantifiers), and proof-level
bindings (eigenvariables). In particular, the combination of Church's
approach to terms and formulas (found in his Simple Theory of Types)
and Gentzen's approach to proofs (found in his sequent calculus)
yields a framework for the interaction of bindings with a full range
of logical connectives and quantifiers. Miller also illustrates how
that framework provides a direct and semantically clean treatment of
computation and reasoning with syntax containing bindings.