Revisiting the proof theory of Classical S4
  In 1965 Dag Prawitz presented an extension of Gentzen-type systems of Natural Deduction to modal concepts of S4. Maria da Paz Medeiros showed in 2006 that the proof of normalisation for classical S4 does not hold and proposed a new proof of normalisation for a logically equivalent system, the system NS4. However two problems in the proof of the critical lemma used by Medeiros in her proof were pointed out by Yuuki Andou in 2009. This paper presents a proof of the critical lemma, resulting in a proof of normalisation for NS4. 
Bruno Lopes, Cec\'ilia Englander, Fernanda Lobo and Marcela Cruz
cs.LO