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.