Section: New Results
Effects in proof theory and programming
Participants : Hugo Herbelin, Gabriel Lewertowski, Étienne Miquey, Alexis Saurin, Matthieu Sozeau.
A classical sequent calculus with dependent types
Dependent types are a key feature of type systems, typically
used in the context of both richly-typed programming languages and
proof assistants. Control operators, which are connected with classical
logic along the proof-as-program correspondence, are known to misbehave
in the presence of dependent types [11], unless
dependencies are restricted to values.
As a step in his work to develop a sequent-calculus version of Hugo Herbelin's
Logical foundations of call-by-need evaluation
Alexis Saurin, in collaboration with Pierre-Marie Pédrot, extended their reconstruction of call-by-need based on linear head reduction with control. They showed how linear head reduction could be adapted to the
Call-by-name forcing for Dependent Type Theory
Guilhem Jaber, Gabriel Lewertowski, Pierre-Marie Pédrot, Matthieu Sozeau, and Nicolas Tabareau studied a variant of the forcing translation for dependent type theory, moving from the call-by-value variant to a call-by-name version which naturally preserves definitional equalities, avoiding the coherence pitfalls of the former one. This new version was inspired by Pierre-Marie Pédrot's former decomposition of forcing in call-by-push-value. It allows to show various metatheoretical results in a succint fashion, notably for the independence of axioms. Work is ongoing to produce more positive results including abstracting reasoning on step-indexing using this technique. This work was presented at LICS 2016 [28].
Classical realizability and implicative algebras
Étienne Miquey has been working with Alexandre Miquel in Montevideo on the
topic of implicative algebras. Implicative algebras are an algebraization of
the structure needed to develop a realizability model.
In particular, they give rise to the usual ordered combinatory algebras
and thus to the triposes used to model classical realizability.
An implicative algebra is given by an implicative structure (which consists
of a complete semi-lattice with a binary operation