Definition ad_monotonic (pf:ad -> ad) :=
    forall a a':ad, Nless a a' = true -> Nless (pf a) (pf a') = true.

Lemma alist_too_low : forall (l:alist A) (a a':ad) (y:A), Nless a a' = true -> alist_sorted_2 ((a', y) :: l) -> alist_semantics A ((a', y) :: l) a = None.