  Usual termination proofs for a functional program require to check all the
possible reduction paths. Due to an exponential gap between the height and size
of such the reduction tree, no naive formalization of termination proofs yields
a connection to the polynomial complexity of the given program. We solve this
problem employing the notion of minimal function graph, a set of pairs of a
term and its normal form, which is defined as the least fixed point of a
monotone operator. We show that termination proofs for programs reducing under
lexicographic path orders (LPOs for short) and polynomially quasi-interpretable
can be optimally performed in a weak fragment of Peano arithmetic. This yields
an alternative proof of the fact that every function computed by an
LPO-terminating, polynomially quasi-interpretable program is computable in
polynomial space. The formalization is indeed optimal since every
polynomial-space computable function can be computed by such a program. The
crucial observation is that inductive definitions of minimal function graphs
under LPO-terminating programs can be approximated with transfinite induction
along LPOs.
