Extensive games are tools largely used in economics to describe decision processes of a community of agents. In this paper we propose a formal presentation based on the proof assistant COQ which focuses mostly on infinite extensive games and their characteristics. COQ proposes a feature called ``dependent types'', which means that the type of an object may depend on the type of its components. For instance, the set of choices or the set of utilities of an agent may depend on the agent herself. Using dependent types, we describe formally a very general class of games and strategy profiles, which corresponds somewhat to what game theorists are used to. We also discuss the notions of infiniteness in game theory and how this can be precisely described.