(* these two lemmas are just negations of the previous ones *)

Lemma inc_dec_not_in_first_out :
forall (q : queue Message) (m : Message) (s3 : Site),
first Message q = value Message m ->
m <> inc_dec s3 ->
~ inc_dec_in_queue s3 q -> ~ inc_dec_in_queue s3 (first_out Message q).