joint work with A.Armando - S. Ranise - In verification with proof assistants (such as PVS, HOL, and Nqthm), decision procedures are typically used for eliminating trivial subgoals represented for instance as sequents modulo a background theory. These theories axiomatize standard data-types such bit-vectors and have proved to be quite useful for, e.g., hardware verification. Elimination of trivial sequents often reduces to the problem of proving the unsatisfiability of conjunctions of literals modulo a background theory , which is the problem we shall consider here. We show how a superposition-based inference system for first-order equational logic can be used almost directly as a satisfiability procedure for various theories including lists, arrays, extensional arrays and combinations of them. We also give a similar decision procedure for homomorphism.