  Ostrogradsky instability generally appears in nondegenerate higher-order
derivative theories and this issue can be resolved by removing any existing
degeneracy present in such theories. We consider an action involving terms that
are at most quadratic in second derivatives of the scalar field and
non-minimally coupled with the curvature tensors. We perform a 3+1
decomposition of the Lagrangian to separate second-order time derivative terms
from rest. This decomposition is useful for checking the degeneracy hidden in
the Lagrangian and helps us find conditions under which Ostrogradsky
instability does not appear. We show that our construction of Lagrangian
resembles that of a GR-like theory for a particular case in the unitary gauge.
As an example, we calculate the equation of motion for the flat FRW. We also
write the action for open and closed cases, free from higher derivatives for a
particular choice derived from imposing degeneracy conditions.
