  e use Prolog as a flexible meta-language to provide executable specifications
of some fundamental mathematical objects and their transformations. In the
process, isomorphisms are unraveled between natural numbers and combinatorial
objects (rooted ordered trees representing hereditarily finite sequences and
rooted ordered binary trees representing G\"odel's System {\bf T} types).
  This paper focuses on an application that can be seen as an unexpected
"paradigm shift": we provide recursive definitions showing that the resulting
representations are directly usable to perform symbolically arbitrary-length
integer computations.
  Besides the theoretically interesting fact of "breaking the
arithmetic/symbolic barrier", the arithmetic operations performed with symbolic
objects like trees or types turn out to be genuinely efficient -- we derive
implementations with asymptotic performance comparable to ordinary bitstring
implementations of arbitrary-length integer arithmetic.
  The source code of the paper, organized as a literate Prolog program, is
available at \url{http://logic.cse.unt.edu/tarau/research/2011/pPAR.pl}
