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

Lemma ad_comp_monotonic :
forall pf pf':ad -> ad,
ad_monotonic pf ->
ad_monotonic pf' -> ad_monotonic (fun a0:ad => pf (pf' a0)).