  In this paper, we present a formalization of Kozen's propositional modal
$\mu$-calculus, in the Calculus of Inductive Constructions. We address several
problematic issues, such as the use of higher-order abstract syntax in
inductive sets in presence of recursive constructors, the encoding of modal
(``proof'') rules and of context sensitive grammars. The encoding can be used
in the \Coq system, providing an experimental computer-aided proof environment
for the interactive development of error-free proofs in the $\mu$-calculus. The
techniques we adopted can be readily ported to other languages and proof
systems featuring similar problematic issues.
