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We perform a Hamiltonian analysis and prove that a large class ofscalar-tensor theories whose Lagrangian is given by a generic function$F\left(K_{ab},N,R_{ab},h^{ab},\nabla_a,t\right)$ is free from the Ostrogradskyghost, and thus these theories contain at most 3 degrees of freedom atclassical level.

Resolving the Ostrogradsky Ghost Problem for a Class of Scalar-tensor Theories