Deterministic bottom-up tree transducers and ground term rewrite systems

We show that it is decidable for any deterministic bottom-up tree transducer A and ground term rewrite system R, which one of the following conditions holds: (i) , (ii) , (iii) , (iv) τ(A) and are incomparable. Here τ(A) is the tree transformation induced by A, and is the congruence generated by R.