Consider the following premises. q_3 or ~s_2. q_3 -> w_4. w -> r. w_4 -> r. s_2 or w. Can we infer ~r from them? If possible, name the inference rules being used at each step.
 From q_3 or ~s_2 we can infer s_2 -> q_3 via rewriting. From s_2 -> q_3, q_3 -> w_4 we can infer s_2 -> w_4 via transitivity. From s_2 -> w_4, w_4 -> r we can infer s_2 -> r via transitivity. Finally, from s_2 -> r, w -> r, s_2 or w we can infer r via disjunction elimination, which contradicts ~r. Therefore, the answer is no.