Consider the following premises. forall x_4: R_2(x_4) -> P_4(x_4). forall x_4: P_4(x_4) or ~P(x_4). forall x_4: ~P_2(x_4) or R_2(x_4). forall x_4: P_2(x_4) or P(x_4). Can we infer forall x_4: P_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) or ~P(x_4) we can infer forall x_4: P(x_4) -> P_4(x_4) via universal rewriting. From forall x_4: P_2(x_4) or P(x_4), forall x_4: ~P_2(x_4) or R_2(x_4) we can infer forall x_4: P(x_4) or R_2(x_4) via universal resolution. Finally, from forall x_4: P(x_4) -> P_4(x_4), forall x_4: R_2(x_4) -> P_4(x_4), forall x_4: P(x_4) or R_2(x_4) we can infer forall x_4: P_4(x_4) via universal disjunction elimination. Therefore, the answer is yes.