Definition SF_cut_down (sf : @SF_seq R) (x : R) :=
  let s := seq_cut_down ((SF_h sf,SF_h sf) :: (SF_t sf)) x in
  mkSF_seq (fst (head (SF_h sf,SF_h sf) s)) (behead s).Definition SF_cut_up (sf : @SF_seq R) (x : R) :=
  let s := seq_cut_up ((SF_h sf,SF_h sf) :: (SF_t sf)) x in
  mkSF_seq (fst (head (SF_h sf,SF_h sf) s)) (behead s).

Lemma SF_cut_down_pointed s x : SF_h s <= x -> pointed_subdiv s -> pointed_subdiv (SF_cut_down s x).