Complete Trigger Selection in Satisfiability modulo first-order Theories

Let T be an SMT solver with no theory solvers except for Quantifier Instantiation. Given a set of first-order clauses S saturated by Resolution (with a valid literal selection function) we show that T is complete if its Trigger function is the same as the literal selection function. So if T halts with a ground model G, then G can be extended to a model in the theory of S. In addition for a suitable ordering, if all maximal literals are selected in each clause, then T will halt on G, so it is a decision procedure for the theory S. Also, for a suitable ordering, if all clauses are Horn, or all clauses are 2SAT, then T solves the theory S in polynomial time.