Definition lfresh h := (last null (get_lows h)) .+ 2.
Definition hfresh h := (last (null .+ 1) (get_highs h)) .+ 2.

Lemma supdomeqUh h1 h2 h :
def (h2 :+ h) -> (def h2 -> supdom h2 h1) -> supdom (h2 :+ h) (h1 :+ h).