  In their paper 'p-adic and real subanalytic sets, J. Denef and L. van den
Dries prove that the theory of the ring of p-adic integers admits the
elimination of quantifiers in the language of p-adic restricted analytic
functions expanded by a division symbol. In this paper, we are interested in
restriction of this language: Let F be any family of restricted analytic
functions, we construct an expansion of F so that the theory of the ring of
p-adic integers is model-complete in the corresponding language. Next, we give
conditions on F so that the model-completeness is effective. Finally, we apply
our results in the context of p-adic exponential rings.
