## Section: Overall Objectives

### Presentation

Mathematics is among the many human activities that have been
transformed by the invention of the computer and its broad diffusion
in the second half of the 20^{th} century. Mathematicians
could, from then on, use a tool allowing to carry out operations
that were too long or too tedious to be executed by hand. Like the
use of the telescope in astronomy, the use of the computer opened
many new prospects in mathematics. One of these prospects is the use
of *proof assistants* , *i.e.* computer programs which
perform some operations on mathematical proofs. The goal of the
research developed in the TypiCal project-team is to develop such
*proof assistants* . The main effort of the project-team is to
contribute to the development proof assistants in general and of the
Coq system in particular,
which
has an important community of users in industry and in academia.
However, we believe that the development of a proof assistant cannot
be accomplished without a joint reflection about the structure of
mathematical proofs and about the use of proof assistants in various
applicative domains.
We also believe that proof assistants should take benefit of the use of
automated deduction tools.
Thus, the questions addressed in the team range
from questions related to the Coq system, such as “What will be the
features of the next version of Coq?”, to more theoretical
questions of logic, such as “What is a proof?” and more applied
ones, such as “How can I delegate part of the proof search to automated
tools?” or “How can we use a proof assistant to check whether a
protocol is free of deadlocks?”.