  Whereas proof assistants based on Higher-Order Logic benefit from external
solvers' automation, those based on Type Theory resist automation and thus
require more expertise. Indeed, the latter use a more expressive logic which is
further away from first-order logic, the logic of most automatic theorem
provers. In this article, we develop a methodology to transform a subset of Coq
goals into first-order statements that can be automatically discharged by
automatic provers. The general idea is to write modular, pairwise independent
transformations and combine them. Each of these eliminates a specific aspect of
Coq logic towards first-order logic. As a proof of concept, we apply this
methodology to a set of simple but crucial transformations which extend the
local context with proven first-order assertions that make Coq definitions and
algebraic types explicit. They allow users of Coq to solve non-trivial goals
automatically. This methodology paves the way towards the definition and
combination of more complex transformations, making Coq more accessible.
