Definition cut (u : Word) :=
  let (a, s) := cut_spec u in (a, let (b, s2) := s in b).Definition cut1 (u : Word) := fst (cut u).Definition cut2 (u : Word) := snd (cut u).

Lemma cut1_Append : forall u v : Word, inmonoid X u -> cut1 (Append u v) = Append u (cut1 v).