Combining ideas coming from Stone duality and Reynolds parametricity, we
formulate in a clean and principled way a notion of profinite lambda-term
which, we show, generalizes at every type the traditional notion of profinite
word coming from automata theory. We start by defining the Stone space of
profinite lambda-terms as a projective limit of finite sets of usual
lambda-terms, considered modulo a notion of equivalence based on the finite
standard model. One main contribution of the paper is to establish that,
somewhat surprisingly, the resulting notion of profinite lambda-term coming
from Stone duality lives in perfect harmony with the principles of Reynolds
parametricity. In addition, we show that the notion of profinite lambda-term is
compositional by constructing a cartesian closed category of profinite
lambda-terms, and we establish that the embedding from lambda-terms modulo
beta-eta-conversion to profinite lambda-terms is faithful using Statman's
finite completeness theorem. Finally, we prove that the traditional Church
encoding of finite words into lambda-terms can be extended to profinite words,
and leads to a homeomorphism between the space of profinite words and the space
of profinite lambda-terms of the corresponding Church type