  The pion-pion scattering amplitudes provided by second-order chiral
perturbation theory are confronted with known rigorous constraints derived from
the axioms of quantum field theory. We mainly test constraints restricting the
$\pi^0$-$\pi^0$ $S$- and $D$-wave amplitudes in the unphysical interval $0\leq
s\leq 4m_\pi^2$. These constraints impose significant lower bounds for a linear
combination of coupling constants specifying the second order chiral
Lagrangian. The accepted value of this combination is consistent with these
bounds. The $\pi^0$-$\pi^0$ $S$- and $D$-waves are strongly correlated by a set
of constraints.
