# Martin Baillon: Talking with trees

Martin Baillon (INRIA, France)

**Abstract:** We generalize to a rich dependent type theory a proof originally developed by Escardó that all System T functionals are continuous. It relies on the definition of a syntactic model of Baclofen Type Theory, a type theory where dependent elimination must be strict, into the Calculus of Inductive Constructions. The model is given by three translations: the axiom translation, that adds an oracle to the context; the branching translation, based on the dialogue monad, turning every type into a tree; and finally, a layer of algebraic binary parametricity, binding together the two translations. In the resulting type theory, every function $f : (\mathbb{N} \to \mathbb{N}) \to \mathbb{N}$ is externally continuous.